Tower of subleading dual BMS charges

We supplement the recently found dual gravitational charges with dual charges for the whole BMS symmetry algebra. Furthermore, we extend the dual charges away from null infinity, defining subleading dual charges. These subleading dual charges complement the subleading BMS charges in the literature and together account for all the Newman-Penrose charges.


Introduction
In recent work [1,2], undertaken with the aim of providing a clear and explicit relation between the asymptotic BMS symmetry and gravitational charges of asymptotically flat spacetimes, we generalised the notion of BMS charges, as defined by Barnich and Troessaert [3], in two complementary ways. The Barnich-Troessaert BMS charges are derived from the general prescription of  for defining asymptotic charges 1 , which in this case turns out to be given by the integral of the Hodge dual of a 2-form H over a 2-sphere at null infinity: 2 where ξ is the asymptotic symmetry generator, g is the background metric and δg is its variation. The variation symbol δ / denotes the fact that the charge is not necessarily integrable. This is generally due to gravitational flux at null infinity.
In Ref. [1], we extended the notion of BMS charges by defining subleading BMS charges as a 1/r expansion of the general prescription of , so that 3 and we showed that the O(1/r 3 ) term gives five of the ten non-linear Newman-Penrose (NP) charges [9]. Writing these charges in the Newman-Penrose formalism [10], we found that these five components, derived from the generalised BMS charge, correspond in some sense to the real part of the NP charges. An obvious question, then, is how do the other five imaginary parts of the NP charges fit into this understanding of generalised BMS charges?
In Ref. [2], inspired by the situation for electromagnetism [11,12], which allows for electric as well as magnetic charges, we defined new dual gravitational charges, associated with supertranslations, as the integral of the Barnich Together, δ /Q 0 and δ /Q 0 can be viewed as the real and imaginary parts of a leading-order supertranslation charge, which can succinctly be written in terms of the leading-order terms 1 There exists an equivalent formalism for defining asymptotic charges, developed by Wald and collaborators [5,6]. Here, we shall continue to work in the framework of the Barnich-Brandt formalism. 2 For an explicit expression for H, see equation (3.3). 3 In principle, assuming analyticity, the tower of charges is infinite, with a charge at each order in the 1/r expansion. However, for physical reasons the metric expansion may only be analytic up to a certain order in 1/r (see e.g. Refs. [7,8]). The tower of BMS charges will then naturally truncate at some corresponding order.
in a 1/r expansion of the Newman-Penrose scalar Ψ 2 , which is a certain null-frame component of the Weyl tensor, and σ, which parametrises the shear of the null congruence of ℓ = ∂/∂r. Thus, there is an attractive correspondence between, on the one hand, the real and imaginary parts of charges written in the complex Newman-Penrose formalism, and on the other hand "electric" and "magnetic" (or "dual") BMS charges definedà la Barnich-Brandt. In this paper, we shall address the problem of generalising the dual charge δ /Q 0 of Ref. [2] to a tower of dual BMS charges (both SL(2, C) and supertranslation charges) as a series in powers of 1/r, in the same manner as the Barnich-Troessaert charge δ /Q 0 was generalised in Ref. [1]. The reason why in Ref. [2] we were able to construct the dual charge at infinity (i.e. at the order 1/r 0 ) by integrating the Barnich We shall construct a tower of bona fide dual gravitational BMS charges as a 1/r expansion away from null infinity, and we shall show that this does, in particular, give rise at the order 1/r 3 to the five imaginary parts of the NP charges. The tower of dual charges is given in terms of a new 2-formH, such that Thus (1.4) gives the same leading-order result that we found in Ref. [2], but now, we are able to extend the construction of dual gravitational charges to the full BMS group and to all subleading orders in a 1/r expansion.
In section 2, we give some preliminary prerequisite information regarding asymptotically flat spacetimes. For a more detailed exposition of the notations and conventions we are using here, the reader is referred to section 2 of Ref. [1]. In section 3, we define the dual gravitational charge corresponding to the full BMS group. We find the full dual BMS charge at leading order in section 4 and investigate the dual charges associated with supertranslations up to order 1/r 3 in a 1/r expansion in section 5. The results of this section are analogous to those obtained for the subleading BMS charges defined in Ref. [1].
Perhaps, most significantly, in section 5.3 we find that the dual charge at order 1/r 3 gives the imaginary parts of the NP charges, something that was missing in the analysis of Ref. [1].
We finish with some comments in section 6.

Preliminaries
We define asymptotically flat spacetimes as a class of spacetimes for which Bondi coordinates (u, r, x I = {θ, φ}) may be defined, such that the metric takes the form [13,14] with the metric functions satisfying the following fall-off conditions 4 at large r F (u, r, where D I is the standard covariant derivative associated with the unit round-sphere metric ω IJ with coordinates x I = {θ, φ} on the 2-sphere. B IJ and C IJ are symmetric tensors with indices raised/lowered with the (inverse) metric on the 2-sphere. Moreover, a residual gauge freedom allows us to require that where h ≡ det(h IJ ) and ω ≡ det(ω IJ ) = sin θ.
We assume, furthermore, that the components T 00 and T 0m of the energy-momentum tensor in the null frame fall off as The Einstein equation then implies that The asymptotic symmetry group corresponding to asymptotically flat spacetimes is the BMS group [13,14], whose corresponding algebra is generated by The Y I are the set of conformal Killing vectors on the round unit 2-sphere, obeying The Abelian part of the algebra, generated by the supertranslations, is (2.11) As will become clear in what follows, it will be convenient to define twisted/dualised objects, as follows: for some symmetric tensor X IJ , we define its trace-free twist/dual bỹ without the need for explicit symmetrisation and we can simply writẽ If X and Y are two symmetric trace-free tensors, then Furthermore, if either one of the symmetric tensors X or Y is trace-free, then

Dual BMS charges
We define the dual BMS charge to be This may be compared with the Barnich-Brandt 2-form H: We foundH by parameterising the most general possible covariant 2-form, built from terms bilinear in ξ and δg and involving one covariant derivative, and determining the constant coefficients by requiring that its integral (3.1) should vanish on-shell when δg is given by the action of an asymptotic symmetry generator, i.e. precisely, the variation of a charge), by demonstrating that δ /Q = 0 when we take δg to be given by (3.4).
Regarded as a 1/r expansion away from null infinity, we have 5 Hence, we find a tower of dual charges, which can be viewed as the charges dual to the BMS charges found in Ref.
In particular, as we shall demonstrate in section 5.4,Ĩ 3 gives the five complementary NP charges [9] that were missing in the analysis of Ref. [1].
We proceed to describe the leading-order dual charge for the full BMS group, before we investigate the subleading terms, corresponding to supertranslations only, in δ /Q in the 1/r expansion given in equation (3.5).

Leading-order dual charges
In this section we derive the leading-order dual BMS charges. These charges ought to be viewed as duals of the Barnich-Troessaert charges found in Ref. [3]. The dual charge as defined in Ref. [2] by taking the integral of the Barnich-Brandt 2-form H (as opposed to the integral of its Hodge dual) cannot incorporate the SL(2, C) part of the BMS group.
However, the dual charge defined in section 3 gives a charge for the full BMS group, as argued for in appendix A.
Using the definition of the asymptotic symmetry generator, given by equation (2.7) and 5 See footnote 3.
the metric coefficients defined in equations (2.2), it is relatively simple to show that Ignoring total derivatives, as these will integrate to zero, and freely integrating by parts, the expression above simplifies tõ Now, using equation (2.9) and the fact that δC IJ is trace-free implies that the order r terms in the expression above vanish, as they should. Moreover, given that which can simply be derived from observing that the symmetric and antisymmetric parts of the expression on the left hand side of the above equation must be proportional to ω IJ and ǫ IJ , respectively, the expression forH θφ in equation (4.2) simplifies tõ whereỸ In summary, we find that This dual charge may be compared with the charge in

Subleading dual charges
In the previous section, we computed the leading-order dual BMS charge for the full BMS group. In this section, for simplicity, we restrict ourselves to the most distinctive part of the BMS group, given by supertranslations, and compute the subleading charges. Thus, hereafter, the generators that will be of interest are those given by equation (2.11).
Furthermore, in this section we require stronger fall-off conditions: Further to the fall-off conditions (2.4), we require which implies These stronger fall-off conditions are needed for the existence of NP charges, whose origin we explain in terms of subleading BMS and dual BMS charges in this section. These conditions allow us to define charges up to order 1/r 3 . In order to define yet further higher order charges we need to impose even stronger fall-off conditions.

Dual charge at O(r 0 )
For the leading-order charge, the contribution of the supertranslations can be simply deduced from the general result (4.6) by turning off the SL(2, C) generators, i.e. the Y I 's.
Hence, from equation (2.8), f = s and charge (4.6) reduces to Moreover, as discussed in Ref. [2], the integrable parts of the two sets of charges may be written as the real and imaginary parts of a single expression 7 More precisely (recalling that s is a real quantity), where, from equations (5.8) and (5.5), respectively,

Dual charge at O(r −1 )
At the next order a simple, if rather tedious, calculation shows that up to total derivatives, which will vanish under integration, Recall (see section 3.2 of Ref. [1]) that assuming the Einstein equation implies that δ /I 1 = 0. (5.14) More generally, without assuming equation (5.13), we may equivalently define

Dual charge at O(r −2 )
A similar long, but simple, calculation finds that Note that this is very similar to δ /I 2 (see equation (3.12) of Ref. [1]). In particular, the integrable part of one is obtained by taking the twist of the tensor fields in the other. The non-integrable part provides an obstruction to the conservation of the integrable charge, and in analogy with the nomenclature adopted for the non-integrable parts of the charges at subleading order in Ref. [1], we may describe such terms here as "twisted fake news." In what follows, we consider whether s can be appropriately chosen such that the twisted fake news vanishes, leaving a conserved/integrable charge. In order to proceed, we need to know how C IJ and D IJ transform under the action of the asymptotic symmetry group; these are given by equations (2.30) and (2.32) of Ref. [1], which we reproduce here for In order to simplify the analysis, we begin by noting that since there is no Einstein equation for F 0 , terms involving F 0 in the non-integrable part of δ /Ĩ 2 , given in the second and third lines of equation ( where, in the second equality, we have used equation (2.15) and, in the third equality, we have used (5.18). For arbitrary F 0 andC IJ (which is trace-free), the above expression vanishes if and only if This is precisely the condition that s is an ℓ = 0 or ℓ = 1 spherical harmonic (see appendix C of Ref. [1]), which implies in particular that Proceeding, while taking s to satisfy equation (5.22), the non-integrable part of δ /Ĩ 2 simplifies which clearly forms a total derivative, and thus vanishes under integration. We therefore conclude that if s is an ℓ = 0 or ℓ = 1 spherical harmonic, However, up to total derivatives and this in fact vanishes upon use of equation (5.22). This analysis is, rather remarkably, completely analogous to that of δ /I 2 in Ref. [1]: the non-integrable part can be made to vanish if and only if s is an ℓ = 0 or ℓ = 1 spherical harmonic, in which case the integrable charge itself turns out to be trivial.
Finally, we express the integrable parts of δ /I 2 and δ /Ĩ 2 as the real and imaginary parts, respectively, of a single charge written in terms of Newman-Penrose scalars. Defining we find that this complex quantity may be written in terms of the integrable charges as

Dual charge at O(r −3 )
Lastly, we consider the dual charge at order 1/r 3 . We find after some algebra that Assuming that T 0m = o(r −5 ), the Einstein equation gives an equation for C I 2 , equation (2.18) of Ref. [1]: Substituting this equation into (5.31) gives we see that the integrable parts are related by replacing the tensor fields in one by the twists of the fields in the other. (Note that trẼ = 0.) As with δ /I 3 , there exist non-integrable terms also, and one may consider, as we did previously at order 1/r 2 , whether there exists some choice of the function s such that the non-integrable part of δ /Ĩ 3 , given by the last three lines of equation ( where the vectors U I and W I can be found by multiplying the above equation by ω JK and ǫ JK , respectively. Only the symmetric part in (JK) is relevant here sinceX IJK is symmetric in (JK). Using the fact that

Discussion
In this paper, we have resolved two puzzles arising from earlier work [1,2]; we have extended the notion of dual gravitational charges to subleading orders in a 1/r expansion away from null infinity and found that at the order 1/r 3 this, together with the subleading charges proposed in Ref. [1], accounts for all ten of the non-linear Newman-Penrose charges.
The tower of dual gravitational charges is constructed from a new 2-formH (see equation On the other hand, we construct the dual charges using a 2-form that is a total derivative (see appendix A). In this sense it is analogous to a magnetic Komar charge and defines a charge without use of the Einstein equation. However, nevertheless we obtain charges that are non-trivial and account for the recently proposed dual gravitational charges [2], extending them to charges associated with the full BMS group, and the imaginary parts of the non-linear Newman-Penrose charges.
Acknowledgements M.G. is partially supported by grant no. 615203 from the European Research Council under the FP7. C.N.P. is partially supported by DOE grant DE-FG02-13ER42020.

A Boundary terms
In this section, we prove that the variation of the dual charge (3.1) vanishes, using the fact that δg ab = 2∇ (a ξ b) .
where we assume an antisymmetrisation in [IJ] on the right hand side for each equality. In the second equality we use the fact that Thus, we conclude that δ /Q as defined in (3.1) vanishes.