Linear Newman-Penrose charges as subleading BMS and dual BMS charges

In this paper, we further develop previous work on asymptotically flat spacetimes and extend subleading BMS and dual BMS charges in a large $r$ expansion to all orders in $r^{-1}$. This forms a complete account of this prescription in relation to the previously discovered Newman-Penrose charges. We provide an explanation for the origin of the infinite tower of linear Newman-Penrose charges with regards to asymptotic symmetries and justify why these charges fail to be conserved at the non-linear level as well as failing to exhibit full supertranslation invariance even at the linear level.


Introduction
In 1968, Newman and Penrose considered a tower of charges that were shown to be conserved in linearised gravity for asymptotically flat spacetimes obeying particular fall-off conditions [1]. In the full non-linear theory, the tower collapses and only the first 10 charges are conserved. Surprisingly, these charges are conserved even in the presence of flux at null infinity [2][3][4]. It is expected that no further such quantities can be constructed that are non-linearly conserved without imposing restrictions on the metric i.e. removing flux at null infinity. It was later shown in Refs. [5,6] that these charges still exist for the broader class of polyhomogeneous spacetimes of [7], with significantly weaker fall-off conditions. It is hence clear that these 10 charges are a prominent feature of asymptotically flat spacetimes with their significance being demonstrated in relation to Aretakis charges on extremal horizons [8][9][10][11].
Recently, the Hamiltonian origin of these charges has been derived [12][13][14][15][16][17] in the covariant phase space formalism [4,18,19] (see also [20][21][22][23][24]). The 10 non-linear charges can be understood as subleading BMS and dual BMS charges for Bondi backgrounds [12,14] (see also [25]), as well as for polyhomogeneous spacetimes [26]. Furthermore, the investigation of [15] also demonstrated that contributions to the Einstein-Hilbert action in the tetrad formalism of general relativity that do not affect the equations of motion may still contain important physics and should not be neglected. In particular, the Holst term [27,28] is trivial by the equations of motion, but gives rise to half of the Newman-Penrose charges as well as the leading order dual charges containing information about the NUT parameter [29,30] and therefore the topology of the spacetime (see for example the recent work of [31]).
The tetrad formalism not only explains the origin of these charges in the context of the asymptotic symmetry group for asymptotically flat spacetimes -the BMS group -but can also be adapted to demonstrate that the charges obey the stronger property of supertranslation invariance, referred to as absolute conservation by Newman and Penrose.
The infinite tower of linear Newman-Penrose charges continues to be a topic of interest (see for example [32]), but is yet to have been investigated in this formalism. This is the purpose of this paper. In Section 2, we begin by reviewing the gauge choices one can make in order to study asymptotically flat spacetimes. We will also choose specific fall-off conditions for the metric functions consistent with those of Newman and Penrose. In Section 3, we obtain the Einstein equations in our gauge and review their structure with regard to initial data. We then obtain the linearised equations at each order in the radial coordinate r −1 .
In Section 4, we review the BMS group and summarise the charge variations obtained in the first order formalism arising from the Palatini and Holst terms. In Section 5, we obtain an expression for the linear Newman-Penrose charges in our gauge choice and demonstrate conservation with respect to u, but the absence of supertranslation invariance. Finally, in Section 6, we demonstrate that the non-linear Newman-Penrose charges, beyond the first 10, are not conserved at any order. We conclude with a brief discussion.
Notation: Latin indices (a, b, ...) denote internal Lorentz indices 0, 1, 2, 3 and are raised and lowered with respect to the Lorentz metric η ab . Indices i, j, ... refer to the 2, 3 components specifically. We use Greek letters (µ, ν, ...) to denote the spacetime indices. The vierbein e a = e a µ dx µ . The spacetime metric can be expressed in terms of the vierbein as ds 2 = g µν dx µ dx ν = η ab e a e b . Indices I, J, ... will denote spacetime indices on the round 2-sphere and will be lowered and raised using the round 2-sphere metric γ IJ and its inverse, respectively, except where explicitly stated otherwise.

Gauge choices
Let (u, r, x I = {θ, φ}) be coordinates on our spacetime manifold such that the metric takes the form [2,3] where r is a radial coordinate. The spacetimes we are considering here are asymptotically flat and we use the Bondi definition of asymptotic flatness where the metric functions obey the following fall-off conditions whereH 0IJ = γ IJ , the round metric on the 2-sphere. Here, we have assumed an analytic expansion of the metric up to a certain order. 1 We define H nIJ ≡H n IJ where , on a 1 N is closely related to the degree of smoothness used by Newman and Penrose.
pair of indices denotes the symmetric trace-free part. Hencē We can use residual gauge freedom to set det h = det γ. The Cayley-Hamilton theorem implies that this is equivalent to requiring where indices are raised and lowered with γ. Plugging the expansion (2.2) into (2.4), contracting with γ IJ and considering the coefficient of r −n , one deduces that TrH n can be expressed in terms ofH i with i = 1, ..., n − 1 and so, in particular, which will be useful for our calculations.
Following Newman and Penrose, we will assume the Newman-Penrose scalar

Null frame
In order to calculate the Einstein equations in a form where they are covariant on the 2-sphere, we require a Lorentz gauge choice. Let our frame fields be e 0 = 1 2 F du + dr, e 1 = e 2β du and e i = rE i I (dx I − C I du), (2.9) where E i I is a zweibein associated with the 2-metric h IJ , so that (h −1 ) IJ E i I E j J = η ij and η ij E i I E j J = h IJ . The inverse of the frame fields are In contrast to all other fields, I, J, ... indices on E i I will be raised and lowered with h IJ . For more details, we refer the reader to Refs. [12,16]. As in the Newman-Penrose formalism, the components of the Einstein tensor can then be written as follows There are different choices for E i I and we will follow Ref. [17] and pick E i I = X I JÊi J with X IJ (u, r, x I ) a symmetric tensor on the 2-sphere. One can express X IJ as an r −1 expansion in terms of H nIJ which means that all Einstein equations are covariant on the 2-sphere.
This choice does not affect our final results.

Einstein equations 3.1 Structure of the Einstein equations and initial data
In this section, we will assume the fall-off conditions of the metric functions but we will generally not need to assume the subleading analytic expansion for the functions in (2.2). In particular, the results below will hold for the polyhomogeneous case considered in Ref. [26]. Throughout, we will assume the following decay of the energy-momentum tensor for a fixed N ≥ 6. As discussed in for example Ref. [12], one cannot assume the components of T ab obey independent fall-off conditions, so in reality, the fall off of T 11 , for example, will typically be much faster. However, any further equations obtained here do not provide new information, so we do not consider them.
As discussed in Ref. [34], the vacuum Einstein equations, in this form, fall into three categories. With the spacetime foliated by u = const hypersurfaces, the G 00 , G 01 and G 0i equations contain no u derivatives and can be viewed as first order differential equations in r for β, F and C I , making them hypersurface equations. The G ij equations are first order differential equations in u and determine how h IJ evolves and are hence viewed as evolution equations. For (2.2), this determines the evolution of H (n≥3)IJ , with H 1IJ unconstrained, constituting free data. The G 11 and G 1i equations are also first order differential equations in u, but are satisfied on r = const hypersurfaces, meaning that if they are true on one such hypersurface, they hold everywhere. These are therefore viewed as conservation equations.
For the metric expansions in (2.2), F 0 and C I 1 cannot be determined from the hypersurface equations, but one can determine their evolution through the conservation equations.
To summarise, one prescribes initial data {F 0 (u 0 , x I ), C I 1 (u 0 , x I ), H (n≥3)IJ (u 0 , x I )} and an arbitrary trace-free tensor H 1IJ (u, x I ) and uses the hypersurface equations to determine all quantities in (2.2) on the hypersurface u = u 0 . Next, the evolution and conservation equations are integrated to find {F 0 , C I 1 , H (n≥3)IJ } at the next time step, with the process then being iterated.
We begin by finding the hypersurface equations. Consider first the 00 component of the Einstein equations. We find that Assuming G 00 = o(r −N ), we then deduce so β can be expressed entirely in terms of h IJ . This is a first order differential equation for β in r with the integration constant changing β up to an O(r 0 ) term, but this contribution is zero by the fall-off condition β = o(r −1 ), so β is completely determined in terms of h IJ .
Next, we compute where D I is the covariant derivative associated with the round 2-sphere metric γ IJ . Assuming G 0i = o(r −N ) and the Einstein equation for β, we can, in principle, write C I as an expression involving only h IJ . 2 This is a second order differential equation in r which means that there are two integration constants in r. The first corresponds to an O(r 0 ) term in the C I expansion, which must be zero by our fall-off condition C I = o(r −1 ). However, the second constant is genuine. For the analytic expansion we consider here, the O(r −3 ) term is undetermined, which means C I 1 is unconstrained for the equations thus far considered, consistent with Ref. [12] . It is also consistent with the results of Ref. [26] for polyhomogeneous spacetimes. As discussed above, this corresponds to initial data and is in fact related to the angular momentum aspect [2,3] Note that E I i is invertible and O(r 0 ) so we deduce the term in brackets in (3.5) is O(r −N ).

Next, we consider
Assuming G 01 = o(r −N ) and the previous two equations, we obtain a first order differential equation for F involving only h IJ and the aforementioned integration constant (C I 1 for our expansion). This, in principle, allows one to determine F up to a term constant in r again. This constant is the coefficient of r −1 in our F expansion, which corresponds to F 0 in (2.2), related to the Bondi mass aspect m = − 1 2 F 0 . This is again consistent with the polyhomogeneous case too. As discussed, this term corresponds to initial data as before.
Importantly, from (3.7), we can deduce that F 0 appears only once in the entire expansion for F , in the r −1 coefficient. That is to say, for n > 0, F n does not depend on F 0 . This can be seen by noticing that capturing the degree of freedom introduced by the integration constant. 3 So far, we have used the hypersurface equations to constrain the metric so that each term in the expansions on some initial hypersurface can only depend on {H (n≥0)IJ , F 0 , C I 1 }. The remaining Einstein equations provide information on how these quantities evolve. We next consider G ij = o(r −(N −1) ). This equation can be simplified for calculation purposes.
First note that the Ricci scalar R = η ab R ab = o(r −(N −1) ), by our fall-off conditions, so that In comparison, to leave G0i invariant in (3.5), we would need C I → C I + δC I , where , contributing at all lower orders.
and is invertible. Furthermore, we will only need to consider the traceless part R IJ as this contains all the new information at each order in r. We find that requiring that This is a first order equation in r for ∂ u h IJ . In principle, it can be solved to produce a first order evolution equation for h IJ . The integration constant from the r equation means that one cannot determine lim r→∞ (r∂ u h IJ ), (3.10) which in our case is ∂ u H 1IJ , constituting free data. More typically denoted as ∂ u C IJ , this is the Bondi news.
The final two equations we consider are the conservation equations G 11 = o(r −2 ) and providing the evolution equations for F 0 and C I 1 respectively in terms of H 1IJ and F 0 where ≡ D I D I is the covariant Laplacian on the unit 2-sphere.

Low order Einstein equations
Now assuming our expansions in (2.2), the equations (3.4), (3.5), (3.7) and (3.9) can be solved order by order in r −1 . For sufficiently large n, it is possible to generalise the results needed in this paper, however, for smaller n, we need to compute the Einstein equations explicitly. Fortunately, this has been done previously in Ref. [12], with the notation We will not reproduce them here and refer the reader to the equations of Ref. [12].

Linear Einstein equations
In this section, we obtain general expressions for the linear terms appearing in Einstein's equations with the fall-off conditions (3.1) assumed. Firstly, considering the expression for D J H IJ n−2 + non-linear terms for n = 3 and 5 ≤ n ≤ N. (3.14) Next, evaluating (3.7) at O(r −n ), one obtains where in the second line we used the C I n−3 equation (3.14) so the equation does not hold for n = 4.
Finally, we look at the O(r −(n−2) ) coefficient in (3.9), where in the second line we have used the equation for C I n−3 in (3.14).

F 0 terms
As discussed in Section 3.1, the F 0 term in (2.2) appears only in the Einstein equations describing the evolution of the initial data {F 0 , C I 1 , H (n≥3)IJ }. In particular, we have with the first two equations immediately following from (3.11) and (3.12). The third equation is obtained by considering the coefficient of O(r −(n−2) ) in (3.9). The F 0 terms in the H n evolution equations are non-linear and we will use these terms later to show that the higher order charges are not conserved for the non-linear theory.

BMS group
We are considering asymptotically flat spacetimes, for which the asymptotic symmetry group is the BMS group [2,35]. This is the set of diffeomorphisms that preserve the form of the metric (2.1) and (2.2). The group is parameterised by supertranslations and conformal Killing vectors on the 2-sphere. As in previous work [12,14,26], we will only focus on the supertranslation part of the BMS group as this is the novel feature. A general generator of such a diffeomorphism is then given by

Subleading charges
It has recently been discovered that there are more charges associated with asymptotically flat spacetimes than previously thought. Charge variations can be derived from different contributions to the action in the tetrad formalism. For a discussion on this, we refer the reader to Ref. [15]. Here, we will focus on the charges arising from the Palatini and Holst Eq. (4.4) can be written covariantly on the 2-sphere as where ǫ IJ is the alternating tensor on the 2-dimensional subspace. We shall consider higher order charges, that is to say, we shall be calculating the above expressions as series in r −1 and considering each coefficient in turn. We write In the following, it will be useful to define the twist of a symmetric tensor X IJ [13,14] If X IJ is trace-free, then X K [I ǫ J]K = 0, so we can drop the symmetrisation in the definition (4.10). Note that the tilde on the quantity H above is not to be confused with the twist of H. The two quantities are a priori unrelated.
Evaluating the charges for our metric (2.1) yields an integrable and non-integrable piece at each order. The non-integrable piece corresponds to flux, with the integrable piece corresponding to a charge that is conserved in the absence of flux [4]. In general, the flux can be made to vanish under physically reasonable conditions on the metric. In Newman and Penrose's derivation of gravitational charges, the metric is allowed to remain general and the flux is made to vanish for a particular choice of spherical harmonic, which recently has been shown to correspond to a particular choice of supertranslation parameter [12].

Linear Newman-Penrose charges
In this section, we obtain expressions for the linear Newman-Penrose charges in this formalism and review their conservation properties. Writing the Weyl tensor as C µνρσ and assuming G 00 = o(r −N ), the first Newman-Penrose Weyl scalar is defined as Considering an expansion of Ψ 0 of the form (N ≥ 5) and evaluating the coefficient of r −(n+5) in (5.1), we find that for 0 ≤ n ≤ N − 5, Defining the differential operators ð andð acting on a scalar η of spin s as [33,37] and noting that Ψ 0 has spin 2, we construct a spin 0 quantity given bȳ where the overall factor is unimportant.

Conservation of linear Newman-Penrose charges
Newman and Penrose demonstrated that their charges are linearly conserved, that is to say their u derivative vanishes without the need to impose constraints on the metric. We shall verify this result in the Bondi gauge. Consider This is zero precisely when s is a linear combination of ℓ = 0, 1 and n − 1 modes. Note that if s is an ℓ = 0 or 1 spherical harmonic, then D I D J s = 0 and the charge is hence trivial, as can be seen from (5.10). Hence, the contribution from such modes can be ignored by the linearity of (5.10) in s. We conclude that s must be a superposition of Y n−1,m modes for m = 0, ±1, ..., ±(n − 1). 4 We now repeat this calculation on the imaginary part of (5.8), again dropping non-linear Now the same argument works as above since H IJ n is an arbitrary symmetric, traceless tensor. Hence, we deduce that (5.8) are conserved and furthermore, they form a basis of all conserved charges that can be constructed in this manner.

Newman-Penrose charges in the tetrad formalism
It has been shown in Refs. [12,14]  Appealing to the lower order Einstein equations in [12] and the linear equations (3.14) and (3.15), the first equation can be re-written, up to total derivatives, as It should be noted that the separation into the integrable and nonintegrable parts is somewhat arbitrary [4]. At linear order, after using Einstein's equations and (4.2), the expressions for the metric variations in terms of s, one obtains The variation is closely related to (5.20) and (5.21) and is given by We start by noting the case in which s 2 = const corresponds to ξ ∼ ∂ u which is the matter considered in the previous subsection. We hence deduce that s 1 must be a superposition of ℓ = n − 1 spherical harmonics for both sets of charges. We shall hence write s 1 ≡ S n−1 where S n−1 = −n(n − 1)S n−1 . For convenience, we we shall drop the subscript on s 2 . Now, up to total derivatives, the expressions above can be re-written This expression can be simplified using the following identity This follows by using the Ricci identity and then the property that S n−1 is an ℓ = n − 1 spherical harmonic. One can then deduce that the final line of (5.26) is zero. Furthermore, the third term can be simplified leaving us with For n = 3, the final two terms clearly vanish for all s. Things are not immediately clear for the first term, but it can be shown that the trace-free symmetric part of the differential operator acting on S n−1 gives zero precisely when acting upon a composition of ℓ = 0, 1 and 2 spherical harmonics 8 , hence (D I D J D K − 1 2 γ K I D J )S 2 = 0. Therefore, we have a non-trivial, supertranslation invariant charge at the linear level. This charge is shown to be non-linearly supertranslation invariant in App. A of [12].
We now focus on n > 3. For X IJ [s; S n−1 ] = 0, it is necessary that for any constant symmetric, trace-free tensor α IJ , S dΩ α IJ X IJ = 0. (5.29) Using the expression for X IJ in (5.28) and integrating by parts, this can be written as 7 Recall that n ≥ 3 so the overall factor is non-zero. 8 This is an adaptation of the arguments in App. C of [12] and App. C of [26] with

Non-linear conservation of Newman-Penrose charges
The full non-linear account of the tetrad formalism for the Palatini and Holst actions has been done in Refs. [12,14] for n ≤ 3. In particular, at O(r 0 ), the BMS and dual charges are discovered and at O(r −3 ), the Newman-Penrose charges are found. Newman and Penrose considered O(r −4 ) and showed that their argument used at the previous order for conservation fails here. In this section, we will present a demonstration of what happens beyond this order by considering specific terms in the non-integrable piece that cannot be made to vanish using the Einstein equations.
We will now demonstrate that the Newman-Penrose charges are not non-linearly conserved for n > 3. In principle, we could begin this discussion as we did in Section 5.1 by settingȲ n+2,m to be general s again; however, this is not necessary. We have learned that in order for the linear terms, involving H IJ n+1 , in ∂ u Q n to vanish, it is necessary for s to be a linear combination of one of the spherical harmonics found by Newman and Penrose. Seeing as there is no Einstein equation relating H IJ n+1 to anything else when it does not appear as a u derivative, there is no chance of the additional non-linear terms possibly cancelling these contributions out. Therefore, for 3 < n ≤ N − 3, the charges must be of the form where S n−1 is a linear combination of ℓ = n − 1 spherical harmonics.
We shall demonstrate that these charges are not conserved by considering terms in ∂ u Q n and ∂ u Q n that contain both F 0 and H IJ n−1 after the use of Einstein's equations. Seeing as there is no Einstein equation for either of these terms (when they do not appear as u derivatives), the ∼ F 0 H IJ n−1 terms need to vanish separately in order for the charge to be conserved. Using (3.9) and (3.17), we find that which is actually valid for 1 ≤ n ≤ N − 3 if one extends the definition of charges to n = 1, 2 using the integrable pieces found in Refs. [12,14]. For n = 1, these terms vanish so there are no F 0 terms at all in Q 1 and Q 1 . For n = 2, we get the non-zero term − 1 4 F 0 C IJ D I D J S 1 9 in Both terms vanish precisely since S 1 is a superposition of ℓ = 0, 1 spherical harmonics, which in both cases make the charge trivial. These results are all consistent with those found in Refs. [12,14]. It is n = 3 where the magic happens. Due to the peeling condition, H 2IJ = 0 so the RHS of (6.3) and (6.4) vanish and there are no such terms in the charge derivatives to potentially destroy conservation. In order to prove these charges really are conserved, one has to consider all terms appearing in the charge derivatives. Fortunately, this has been done in [1] (see also [12,14]). For n > 3, for general H nIJ and F 0 , both (6.3) and (6.4) are non-zero and hence the charges cannot possibly be conserved. The introduction of this non-linear term alone ruins the conservation of the higher order Newman-Penrose charges.
It is unusual that the peeling condition is what protects the 10 non-linearly conserved Newman-Penrose charges given it has been shown that relaxing the peeling condition still gives rise to conserved charges at this order in r. For a more detailed discussion on this in this formalism, we refer the reader to Ref. [26]. This analysis justifies Newman and Penrose's claim that there are no further charges of this form. Seeing as the charges are not conserved in u for n > 3, it follows immediately that the charges are not supertranslation invariant. 10

Discussion
In this paper, we have further investigated whether or not there are further Newman-Penrose charges beyond the existing 10. This can be seen as a step towards the conclusion of the investigation carried out in Refs. [12] and [14], extending the arguments to all orders in r −1 within the realm of what is mathematically viable. We verified that the linear Newman-Penrose charges are indeed conserved with respect to u in this formalism, as ought to have 9 Recall that the more standard notation is H IJ 1 ≡ C IJ . 10 Recall that the supertranslation with parameter s = const is in fact ξ ∝ ∂u. The charge is invariant under this paramter iff it is conserved in u.
been the case given Newman and Penrose's findings. However, the quantities are not charges in the sense of the tetrad formalism and in particular, do not exhibit supertranslation invariance. It has already been shown in Ref. [12] that the 10 non-linear Newman-Penrose charges are fully supertranslation invariant, as was also shown by Newman and Penrose.
Although hope is lost in showing that the linear Newman-Penrose charges have a place in this formalism, there remains several other possibilities that one can argue. It is possible that the non-integrable pieces can be made to vanish by instead restricting the metric in the same fashion as one does with the BMS and leading order dual BMS charges. These charges exist for any s and are conserved in the absence of Bondi news, i.e. ∂ u C IJ = 0. It is possible that imposing similar constraints on the metric may result in the non-integrable piece vanishing at some other order of r −1 , however, seeing as there is no Einstein equation relating H nIJ to other terms (except when it appears as a u derivative), the arguments in this paper regarding the linear terms in the non-integrable piece still hold and we would require conditions on the metric such as for which it is difficult to find physical meaning, not to mention that these would be a subset of many further requirements when one considers the entire non-integrable piece.
Another possibility is that the separation into the integrable and non-integrable piece may be adjusted. Perhaps by moving terms from the non-integrable piece into the integrable piece, it is possible to remove the F 0 terms in such a way that the non-integrable piece can be made to vanish for some s. Given the rapidly increasing complexity of the equations at each order, it is difficult to believe that this will work, but it cannot be ruled out without having a clearer understanding of how one chooses the integrable piece in this formalism, a problem that is still to be addressed at subleading orders.
In summary, we have shown that although the infinite tower of non-linearly conserved Newman-Penrose charges do appear very naturally in the tetrad formalism, they are not charges in the traditional sense. That being said, one can argue that the charges are still of mathematical significance. In particular in [38], the infinite tower is realised with a different origin, motivated by gravitational memory effects [39][40][41][42].
Since their discovery, the charges' conservation properties have been shown to hold true in even broader classes of spacetimes with slower fall-off conditions on the Weyl scalar [6,26].
The Newman-Penrose charges are a prominent feature of asymptotically flat spacetimes and this study further demonstrates their significance through their individuality.

Acknowledgements
I would like to thank Mahdi Godazgar for discussions that initiated this study and his support throughout this project. I am supported by a Royal Society Enhancement Award.

A Supertranslation invariance of BMS and dual charges
We demonstrate the supertranslation invariance of the BMS and dual BMS charges, which in turn implies their conservation in u. We will write H 1IJ ≡ C IJ throughout. There is a pair of charges for each spherical harmonic parameterised by ℓ ≥ 0 and m = 0, ±1, .., ±ℓ,