Finite BMS transformations

The action of finite BMS and Weyl transformations on the gravitational data at null infinity is worked out in three and four dimensions in the case of an arbitrary conformal factor for the boundary metric induced on Scri.


Introduction
There are two main applications of two dimensional conformal invariance [1]. The first consists in using Ward identities associated to infinitesimal symmetry transformations in order to constrain correlation functions. In the second application, starting from known quantities in a given domain, the finite transformations are used to generate the corresponding quantities pertaining to the transformed domain (see e.g. [2]). In this case, the Schwarzian derivative occuring in the transformation law of the energy-momentum tensor plays a crucial role.
The aim of the present paper is to derive the finite transformations necessary for the second application. In particular for instance, if one knows the theory in the form of an asymptotic solution to classical general relativity for the standard topology S 2ˆR of I`, one can use the transformation laws to get the solution on a cylinder times a line. Particular aspects of such mappings in general relativity have been discussed previously for instance in [15][16][17]. More concretely, in the present paper we will work out the transformation laws of asymptotic solution space and the analog of the Schwarzian derivative for finite extended BMS 4 transformations and local time-dependent complex Weyl rescalings. Whereas the former corresponds to the residual symmetry group, the latter represents the natural ambiguity in the definition of asymptotically flat spacetimes in terms of conformal compactifications [18,19].
As a warm-up, we start by re-deriving the known finite transformations in three dimensions in the asymptotically anti-de Sitter and flat cases. In the former case, one recovers the Schwarzian derivative as an application of the AdS 3 /CFT 2 correspondence [20,21]. In the latter case, one obtains the finite transformation laws for the Bondi mass and angular momentum aspects that have been previously obtained by directly integrating the infinitesimal transformations [22]. In both these three dimensional cases, these results are generalized to include local Weyl transformations. In other words, we are working out the action of finite Penrose-Brown-Henneaux transformations in the terminology of [23,24].
To summarize the results for the simplest case when computations are done with respect to the Riemann sphere, i.e., when the metric on I`is taken as ds 2 " 0dr u 2´2 dζdζ, the extended BMS 4 group consists of superrotations ζ " ζpζ 1 q, ζ " ζpζ 1 q together with supertranslations r u 1 " p Bζ Bζ 1 Bζ Bζ 1 q´1 2 rr u`βpζ, ζqs. In particular, the asymptotic part of the shear, the news, and the Bondi mass aspect transform as where t¨,¨u denotes the Schwarzian derivative.

Adapted Cartan formulation
In the Cartan formulation of general relativity, the fundamental fields are on the one hand, a vielbein, e a µ , together with its inverse e a µ and associated metric g µν " e a µ η ab e b ν , where η ab is constant and, on the other hand, a Lorentz connection satisfying the metricity condition ∇ a η bc " 0, Γ abc " η ad Γ d bc " Γ rabsc . Indices are lowered and raised with η ab and g µν and their inverses. The associated connection 1-form is Γ a b " Γ a bc e c with e c " e c µ dx µ . The torsion and curvature 2-forms are given by T a " de a`Γa Bx ν¯p xq, where the last expression is equivalent to the transformation law for the connection 1- Bx µ . Equations of motion deriving from the variational principle Sre, Γs " 1 16πG ż d d x epR abcd η ac η bd´2 Λq, (2.2) are equivalent to T a " de a`Γa b e b " 0 and Einstein's equations, G ab`Λ η ab " 0. Together with the metricity condition, the former implies where the structure functions are defined by D c ab e c " pe a pe b µ q´e b pe a µ qq B Bx µ . Conversely, T a " 0 is equivalent to D cab "´2Γ crabs .
In order to describe Lorentz transformations, one associates to a real vector v " v a e a a 2ˆ2 symmetric matrixv " v a p j a , where p j a are chosen as where ǫ "˜0 1 1 0¸. (3.7) For g P SLp2, Rq, one considers the transformation with ad´bc " 1 and a, b, c, d P R, then where the first index is the lign index. SLp2, Rq group elements will be parametrized as Using ω a " 1 2 ǫ abc Γ bc and the transformation law of the Lorentz connection given below (2.1), we have In terms ofω "ĵ a ω a , this is equivalent tô Explicitly, for the spin coefficients encoded in one findsω 1 a " Λ a c gω c g T´g ǫe 1 a pg T q. (3.15) In this case, Einstein's equations are equivalent to Alternatively, one can use q v "vǫ in order to describe real vectors by traceless 2ˆ2 matrices. The associated basis is In this case, we have and dq e´2q ωq e " 0, dq ω´q ωq ω´Λ 2 q eq e " 0.

Fefferman-Graham solution space
In the AdS 3 case, Λ "´L´2 ‰ 0, we start by rederiving the general solution to the equations of motion in the context of the Newman-Penrose formalism. We will recover the on-shell bulk metric of [35], but with an arbitrary conformal factor for the boundary metric [21] (see also Section 2 of [7] in the current context).
The analog of the Fefferman-Graham gauge fixing is to assume that which is equivalent to Γ ab3 " 0 and can be achieved by a local Lorentz transformation. This means that the triad is parallely transported along m and that m is the generator of an affinely parametrized spatial geodesic. In this case, ∇ ra m bs " n ra l bs pπ`τ q so that m is hypersurface orthonormal if and only it is a gradient, which in turn is equivalent to This condition will also be imposed in the following.
Introducing coordinates x µ " px`, x´, ρq, µ " 1, 2, 3 such that m is normal to the surfaces ρ " cte and the coordinate ρ is the suitably normalized affine parameter on the geodesic generated by m, the triad takes the form where a " p`,´q. The associated cotriad is where ǫ`´" 1 "´ǫ´`and ǫ˘˘" 0. In order to compare with the general solution given in [7], one introduces an alternative radial coordinate r " e ρ ?
2L , in terms of which Under these assumptions, the Newman-Penrose field equations (A.1)-(A.12) can be solved exactly. Indeed, the three equations (A.1), (A.7) and (A.9) reduce to the system which is solved by introducing the complex combinations L˘" τ˘i ? νκ. The general solution is given by (4.7) The last two radial equations involving the spin coefficients, equations (A.3) and (A.8), simplify to δǫ " τ ǫ`κγ, δγ " τ γ´νǫ, (4.8) and are solved through The last radial equations are (A.11) and (A.12). Their r-component are trivially satisfied while their components along x˘are of the same form than (4.8), δl˘" τ l˘`κn˘, δn˘" τ n˘´νl˘, (4.10) which leads to In these equations, C i , K1 , K2 are functions of x a " x˘alone.
Note that asymptotic invertibility of the triad is controlled by the invertibility of the matrix formed by these functions, Using the radial form of the various quantities, equations (A.2) and (A.6) are equivalent to which then implies that equation (A.4) reduces to while the components along x˘of equation (A.10) become Because of invertibility of the matrix (4.12), equations (4.15) and (4.14) can be used to express C 4 , C 5 and C 1 in terms of K a 1 and K a 2 . The two equations in (4.13) then become dynamical equations for C 2 and C 3 . Since we now have treated all Newman-Penrose equations, the solution space is parametrized by K a 1 , K a 2 and by initial conditions for C 2 and C 3 .
In the limit r going to infinity, the triad elements l and n given in (4.11) take the form l˘" r´1K1`Opr´3q, n˘" r´1K2`Opr´3q. With a change of coordinates on the cylinder, we can make the associated asymptotic metric explicitly conformally flat. This amounts to the choice Introducing this into equations (4.14) and (4.15), we get while the dynamical equations (4.13) reduce to B`C 2`2 B`ϕC 2 " L 2 B´`e´2 ϕ B´B`ϕ˘, With the extra conditions (4.16), the space of solutions is parametrised by three functions ϕ, C 2 and C 3 defined on the cylinder with coordinates x˘such that equations (4.18) are valid. These two equations can be integrated directly but we will derive the explicit form of C 2 and C 3 in a different way using the action of the asymptotic symmetry group below.

Residual gauge symmetries
The residual gauge transformations are the finite gauge transformations that preserve the set of asymptotic solutions. Since these transformations map solutions to solutions, once the conditions that determine the asymptotic solution space are preserved, no further restrictions can arise. A gauge transformation is a combination of a local Lorentz transformation and a change of coordinates of the form The unknowns are A, B, E, r and x˘as functions of r 1 , x 1˘.
Using the a " 3 component of the transformation law of the triad, the requirement m 1µ " r 1 ? 2L δ µ r 1 is equivalent to Expanding for each coordinate, we get 2L . In order to implement the gauge fixing condition on the new spin coefficients q ω 1 3 " 0, we first rewrite the last equation of (3.19) as When a " 3 this becomes 24) and is equivalent to three conditions on the Lorentz parameters, When suitably combining these equations, one finds The set of equations (4.22) and (4.26) forms a system of differential equations for the radial dependence of the unknown functions. In order to solve it asymptotically, we will assume that the functions have the following asymptotic behavior, (4.27) Inserting this into the equations, we easily get where we have assumed r r 1 ą 0 asymptotically. At this stage, we have fixed the radial dependence of all the unknown functions and we are left with six functions A 0 , B 0 , E 0 , r 0 , x0 of x 1˘.
We now have to require m 1 µ " ? 2L r 1 δ r 1 µ . However, since we have inmposed m 1 " ? we then get The last condition we have to require is the asymptotically conformally flat form of the new triad. This can be done by imposing The leading terms of e 11 ν " Λ 1 a e a µ Bx µ Bx 1ν and e 12 Combining with equation (4.30), allows one to extract A 0 , B 0 , E 0 in terms of the other functions, It thus follows that the residual gauge symmetries are determined (i) by the change of variables x˘" x0 px 1˘q at infinity, each depending on a single variable, which we assume to be orientation preserving B 1 x0 ą 0 ă B 1 x0 , and (ii) by r 0 px 1`, x 1´q .
For notational simplicity, we drop the subscript 0 on the change of variables at infinity and on the Weyl parameter in the next section.

Action of conformal and Weyl group
The group obtained in the previous section is the combined conformal and Weyl group and is parametrized by´x The last equation of (4.32) encodes the transformation law of ϕ, Note that, as a consequence, if rpx 1`, x 1´q , r s px 2`, x 2´q and r c px 2`, x 2´q are associated to a first, a second successive and the combined transformation respectively, the composition law is This group reduces to the conformal group for fixed conformal factor of the boundary metric: when ϕ " ϕ 1 it follows from (4.35) that r is determined by the change of variables at infinity, r "´1 2 ln pB 1 x`B 1 x´q. When freezing the coordinate transformations, one remains with the additive group of Weyl rescalings that amount here to arbitrary shifts of ϕ.
As discussed in Section 4.1, the on-shell metric, triads and spin connections are entirely determined by the arbitrary conformal factor ϕpx`, x´q and the integration functions C 2 px`, x´q, C 3 px`, x´q satisfying (4.18). To obtain the action of the group on the latter, we can extract the subleading terms of l 1 px`q " Λ b 1 eb and n 1 px´q " Λ b 2 eb . This gives which can also be written in terms of ϕ using equation (4.35). Note that, by construction, the transformed C 1 2 px 1`, x 1´q , C 3 px 1`, x 1´q have to satisfy the transformed equations, i.e., equations (4.18) where all quantities, C 2 , C 3 , ϕ, x˘, B˘are primed.
In the particular case where ϕ " 0, equations (4.18) reduce to B`C 2R " 0, B´C 3R " 0 so that C 2R " p8πGLqT´´px´q and C 3R " p8πGLqT``px`q. Applying the particular Weyl transformation x 1˘" x˘, r " ϕ 1 , and removing all primes, we obtain from (4.37) that the general solution to the dynamical equations (4.18) for arbitrary ϕ is given by (4.38) Solution space can thus also be parametrized by the conformal factor ϕ and the two integration functions T˘˘px˘q depending on a single variable each. The action of the asymptotic symmetry group on the latter can be extracted from equations (4.37), in terms of the Schwarzian derivative for a function F of x, and with the characteristic values of the central charges for asymptotically AdS 3 gravity [36]. In other words, the integration functions T˘˘are Weyl invariant, while under the centrally extended conformal group, one recovers the well-known coadjoint action, i.e., the standard transformation law of an energy-momentum tensor.

Solution space
The first gauge fixing conditions that we will assume are This is equivalent to Γ ab1 " 0 which can be achieved by a suitable Lorentz rotation. It implies that the tetrad is parallely transported along l and that l is the generator of an affinely parametrized null geodesic. In this case, ∇ ra l bs "´2l ra m bs pτ´βq, so that l is always hypersurface orthornormal. It is a gradient if and only if a condition which will also be imposed in the following.
Introducing Bondi coordinates x µ " pu, r, φq, µ " 0, 1, 2 such that the surfaces u " cte are null with normal vector l, l µ " δ 0 µ and such that r is the suitably normalized affine parameter on the null geodesics generated by l, the triad takes the form The associated cotriad is Under these assumptions, the Newman-Penrose equations (A.1)-(A.6) fix the r dependence of all spin coefficients according to When used in equations (A.10) and (A.11), the r dependence of the triad is In order to solve the remaining equations, we will assume in addition that The first condition can be satsified by changing the affine parameter r Ñ r`C 1 . We can then do a Lorentz transformation with a " d " 1, c " 0 and b " C 2 in order to impose C 2 " 0, and finally a change of coordinates B u φ 1 "´K 2 B φ φ 1 to obtain K 2 " 0. Note however that both of these last two transformations are only valid asymptotically. Requiring them to preserve the gauge fixing conditions will require subleading terms in a similar way as in the computation of section 5.2. On the level of solutions, the additional conditions simply amount to setting Redefining K 1 " e´ϕ, the remaining equations, i.e., (A.7)-(A.9) and (A.12), are equivalent to These equations can be integrated directly, but we will again generate the solution by using the asymptotic symmetry group below.

Residual gauge symmetries
The residual gauge symmetries again consist of the subset of gauge transformations that preserve the set of conditions determining the asymptotic solution space. We will consider a general change of coordinates of the form combined with an arbitrary local Lorentz transformation. The unknowns are A, B, E, u, r, φ as functions of u 1 , r 1 , φ 1 .
Using the a " 1 component of the transformation law for the triad it follows that imposing l 1µ " δ µ r 1 is equivalent to the radial equations, Bu The gauge fixing on the new spin coefficients takes the form q ω 1 1 " 0. The component a " 1 of the last equation of (3.19) can be rewritten as This is equivalent to three conditions on the rotation parameters, which can be suitably combined to yield The set of equations (5.14) and (5.16) forms a system of differential equations for the radial dependence of the unknown functions. In order to solve it asymptotically, we assume that the functions have the following asymptotic behavior, The unknown r can be traded for χ " re´E " Opr 1 q satisfying The solution is given by χ " r 1`χ 0 pu, φq`Opr 1´1 q, (5.19) which, when introduced into the other radial equations, gives At this stage, we have fixed the radial dependence of all the unknowns and are left with six functions A 0 , B 0 , E 0 , u 0 , φ 0 , χ 0 of u 1 and φ 1 .
We now have to require l 1 µ " δ u 1 µ . After having imposed l 1 " B Br 1 , one has in particular that l 1 r 1 " 0. This follows from the combination of l 1 For the remaining components of l 1 µ it is enough to verify that l 1 Extracting the leading order from we get We still have to impose three conditions: The first one is a condition on the triad and can be imposed by requiring e 13 u " Op1q. More generally, we have and, requiring the new cotriads to have the same form in the new coordinate system than they had in the old one, the leading terms of e 13 u and e 13 φ yield Note in particular that our choice of parametrization for the Lorentz rotations leads to Bφ 0 Bφ 1 ą 0. The first equation is equivalent to V 1 " Opr 1´1 q while the second one gives the transformation law of ϕ. Combining (5.23) with (5.21), we obtain To implement the last two conditions, we will use the transformation law of q ω 3 given in the last equation of (3.19). Imposing pq ω 1 3 q 11 " Opr 1´2 q and pq ω 1 From the general solution, we have Inserting this into the two equations we can extract the value of A 0 and χ 0 , The asymptotic symmetry group is thus parametrised by three functions u 0 pu 1 , φ 1 q, φ 0 pu 1 , φ 1 q and E 0 pu 1 , φ 1 q satisfying the constraints Note that, when taking these into account, the Jacobian matrices for the change of coordinates at infinity arë For notational simplicity, we will drop the subscript 0 on the functions determining the change of coordinates at infinity and on the Weyl parameter in the next two section.

Combined BMS3 and Weyl group
From equation (5.27), it follows that Epu 1 , φ 1 q is determined by the function upu 1 , φ 1 q and, conversely, that the knowledge of such a function E allows one to recover the complete change of coordinates, up to an arbitrary functionû 1 pφ 1 q, Note that the point with coordinates pû 1 pφ 1 q, φ 1 q in the new coordinate system is described by p0, φq in the original coordinate system. When considering the inverse transformation, we can write, where E 0 is now considered as a function of the original coordinate system through E 0 pu 1 pu, φq, φ 1 pu, φqq and the point with coordinates pûpφq, φq is described by p0, φ 1 q in the new coordinate system.
Alternatively, one can define Upu, φq " e ϕ r u, αpu, φq " e ϕ β, Note that if ϕ does not depend on u then r U " e´ϕu, β "´e´ϕûpφq whereas U " u and αpφq "´û. If furthermore ϕ 1 does not depend on u 1 , then neither does E and u 1 pu, φq " e Epφ 1 q pu`αq. The standard definition of the BMS 3 group is then recovered when the conformal factor is fixed to be zero, i.e., when ϕpu, φq " 0 " ϕ 1 pu 1 , φ 1 q, in which case it follows from equation (5.31) that the Weyl transformations are frozen to

Action on solution space
Solution space is parametrized by the three functions ϕ, K 3 , K 4 satisfying the evolution equations (5.10). The action of the group on the conformal factor ϕ has already been computed in the previous section. We can extract the transformation law of K 4 from pq ω 1 3 q 12 and the one of K 3 from the second order of e 11 φ , In the particular case where ϕ " 0, equations (5.10) reduce to B u K 4R " 0, B u K 3R " B φ K 4R , so that K 4R " p16πGqppφq, K 3R " p16πGqpjpφq`uB φ pq. Applying the particular Weyl transformation φ 1 " φ, u 1 " ş u 0 dv e E , Epu 1 , φ 1 q " ϕ 1 pu 1 , φ 1 q, with inverse transformation u " ş u 1 0 dv 1 e´ϕ 1 pv 1 ,φ 1 q , we obtain from (5.40) that After removing all primes and writing the inverse transformation as in (5.34), it follows that the general solution to the dynamical equations (5.10) for arbitray ϕ is given by The final parametrisation of the solution space studied in section 5.1 is given by the conformal factor ϕ and the two functions ppφq and jpφq. Their transformation laws under the combined BMS 3 and Weyl group is given by The central charges have the characteristic values for asymptotically flat three-dimensional Einstein gravity [37]. These quantities are thus Weyl invariant, which needs to be the case by construction since a Weyl transformation applied to K 3 , K 4 amounts to applying the combined Weyl transformation to p, j with the associated change of r u. Their transformations under the BMS 3 group agree with those derived by different methods in [22,38].
In order to describe Lorentz transformations in four dimensions in terms of a null tetrad, one associates to a real vector v " v a e a , with v 1 , v 2 P R, v 4 " v 3 P C, a 2ˆ2 hermitian matrix p v " v a p j a , where the p j a are chosen as In this case For an element g P SLp2, Cq, one considers the transformation More explicitly, if where the first index is the lign index.
The standard three classes of rotations [30] are then given by • class I for which l 1 " l, m 1 " m`Al, m 1 " m`Al, n 1 " n`Am`Am`AAl: a " 1 " d, c " 0, b "´A, A P C, • class II for which n 1 " n, m 1 " m`Bn, m 1 " m`Bn, l 1 " l`Bm`Bm`BBn: a " 1 " d, b " 0, c "´B, B P C, • class III for which l 1 " e´E R l, n 1 " e E R n, m 1 " e iE I m, m 1 " e´i E I m: Finally, the SLp2, Cq group element corresponding to a combined rotation II˝I˝III is given by the transformation law of Lorentz connection becomes p ω 1 " gp ωg T´g ǫdg T . (6.12) More explicitly, for the spin coefficients encoded in p ω 1 "˜π´ǫ ǫ κ¸, p ω 2 "˜ν´γ γ τ¸, p ω 3 "˜µ´β β σ¸, p ω 4 "˜λ´ά α ρ¸, (6.13) one finds p ω 1 a " Λ a c gp ω c g T´g ǫe 1 a pg T q. (6.14) Alternatively, one can use q v " p vǫ in order to describe real vectors. The associated basis is where q ω 1 "˜ǫ π κ´ǫ¸, q ω 2 "˜γ ν τ´γ¸, q ω 3 "˜β μ σ´β¸, q ω 4 "˜α λ ρ´α¸. (6.18) For the Weyl scalars, we follow the conventions of [33,39], which differ by a sign from those of [30] and those of [25,27] (when taking into account in addition the correction for Ψ 2 given in [26,29]). If C abcd denote the components of the Weyl tensor and Ψ ABCD the associated Weyl spinor, Their transformations law under Lorentz rotations can be either worked out directly by using C 1 a 1 a 2 a 3 a 4 " Λ a 1 b 1 . . . Λ a 4 b 4 C b 1 b 2 b 3 b 4 and the symmetries of the Weyl tensor, as done in [30] for the individual rotations of type I, II, III. A faster way is to use the correspondence with the Weyl spinor: with our choice of Infeld-van der Waerden symbols in (6.4) (cf. (3.1.50) of [39]), e a " pl, n, m, mq Ø ǫ A ǫ 9 A " pǫ 0 ǫ 0 , ǫ 1 ǫ 1 , ǫ 0 ǫ 1 , ǫ 1 ǫ 0 q, e 1 a " Λ a b e b with Λ a b as in (6.9) corresponds to ǫ 1 When taking into account the complete symmetry of the Weyl spinor, one gets directly from Ψ 1

Newman-Unti solution space
The gauge fixing conditions at null infinity 1 that are usually assumed correspond to imposing the six real conditions encoded in κ " ǫ " π " 0. This is equivalent to requiring Γ ab1 " 0 and can be achieved by a suitable Lorentz rotation. According to the definition of the Newman-Penrose scalars, it implies that the whole tetrad is parallely transported along l, Dl " 0 " Dn " Dm " Dm. In particular, this means that l is the generator of affinely parametrized null geodesics. One then requires in addition that l is hypersurface orthonormal and a gradient, which yields 3 more conditions, ρ " ρ and τ " α`β, see, e.g., section 1.9 of [30].
This allows one to choose Bondi coordinates x µ " pu, r, x A q, µ " 0, . . . , 3, A " 2, 3, x A " pζ, ζq such that the surfaces u " cte are null with normal vector l, l µ " δ 0 µ and that r is the suitably normalized affine parameter on the null geodesics generated by l. The tetrad then takes the form Note furthermore that if ξ A " g AB ξ B with g AB the two dimensional metric inverse to g AB , then ξ Aξ A "´1, ξ A ξ A " 0 "ξ Aξ A . The associated cotetrad is given by On a space-like cut of I`, we use coordinates ζ, ζ, and the metric ds 2 "´γ AB dx A dx B "´2pP P q´1dζdζ, (6.24) with P P ą 0. For the unit sphere, we have ζ " cot θ 2 e iφ in terms of standard spherical coordinates and P S pζ, ζq " The covariant derivative on the 2 surface is then encoded in the operator ðη s " P P´sBpP s η s q, ðη s " P P s BpP´sη s q , (6.26) where ð, ð raise respectively lower the spin weight by one unit. The weights of the various quantities used here are given in table 1. Complex conjugation transforms the spin weight into its opposite and leaves the conformal weight unchanged. Note that P is of spin weight 1 and "holomorphic", ðP " 0 and that rð, ðsη s " s 2 R η s , (6.27) with R " 2P P BB lnpP P q " 2ðð lnpP P q, R S " 2. We also have rB u , ðsη s " pB u ln P ð`sðB u ln P qη s . (6.28) According to [25][26][27], once the conditions κ " ǫ " π " 0 are fixed and coordinates u, r, ζ, ζ such that l ν " δ u ν , l ν " δ ν r are chosen, which implies in particular also that ρ´ρ " 0 " τ´α´β, the leading part of the asymptotic behaviour given in (6.29) follows from the equations of motion, the condition Ψ 0 " Ψ 0 0 r´5`Opr´6q and uniform smoothness, i.e., a standard restriction on the functional space imposing how the fall-off conditions in r behave with respect to differentiation. In addition, the choice of a suitable radial coordinate is used to put to zero the term in ρ of order r´2, while by a choice of coordinates x A , the leading part r 2 of the spatial metric is set to be conformally flat, and the constant part of X A to vanish. Finally, the leading order r´1 of τ is set to zero by a suitable null rotation. As will be explicitly seen below, these conditions guarantee that the asymptotic symmetry group is the extended BMS group combined with complex rescalings.
For the explicit form of asymptotic solution space, we will follow closely [27] (see also [25,26]), except that the complex P used here is twice the P used there and the ð operator is taken to agree with the definition used in [33]. Furthermore, ζ " x 3`i x 4 and ∇ " 2B. More details can be found for instance in the reviews [29,31,33,39] and also in [40], where a translation to results in the BMS gauge as used in [7] can be found. Note also that, as compared to [7,9,10,40], we have changed the signature of the metric in order to agree with the standard conventions used in the context of the Newman-Penrose formalism and that x 4 Ñ´x 4 .
The asymptotic expansion of on-shell spin coefficients, tetrads and the associated components of the Weyl tensor are given by and In this approach to the characteristic initial value problem, freely specifiable initial data at fixed u 0 is given by Ψ 0 pu 0 , r, ζ, ζq in the bulk (with the assumed asymptotics given above) and by pΨ 0 2`Ψ 0 2 qpu 0 , ζ, ζq, Ψ 0 1 pu 0 , ζ, ζq at I`. The news σ 0 pu, ζ, ζq is free data at I`for all u and determines, together with the other initial data at I`, the would-be conserved BMS currents.
As in [41] (see also [42]), for a field η s,w of spin and conformal weights ps, wq, one can associate a field r η h,h of conformal dimensions ph,hq through η s,w " P h Phr η h,h , ph,hq "`´1 2 ps`wq, The conformal dimensions of the various quantities used here are given in table 2.
When expressed in these quantities, (6.33) and (6.34) become and Below, during the construction of the solution to the evolution equations (6.33) and (6.34), we will construct improved fields of this type that take due care both of the additional udependence and of the inhomogeneous parts of the transformation laws.

Residual gauge symmetries
The residual gauge symmetries are the combined Lorentz transformations and coordinate changes that leave on-shell spin coefficients and tetrads invariant. Since these transformations map solutions to solutions, once the conditions that determine asymptotic solution space are preserved, no further restrictions can arise. The change of coordinates is of the form it follows that imposing l 1µ " δ µ r 1 is equivalent to replacing the LHS by Br 1 " ddl ν`c cn ν´p dcm ν`c .c.q, or more explicitly, ‰ .

(6.40)
In order to implement the gauge fixing conditions in the new coordinate system, or equivalently q ω 1 1 " 0, we rewrite the last of (6.17) as and require, for a " 1, g´1 Bg More explicitly, the conditions on the rotation parameters are Note that the additional equation involving Λ 1 b pq ω b q 22 "´Λ 1 b pq ω b q 11 on the RHS follows from the first equation when using ad´bc " 1. When suitably combining these equations, one finds (6.43) The set of equations (6.40) and (6.43) forms a system of differential equations for the radial dependence of the unknown functions. In order to solve it asymptotically, we assume that the functions have the following asymptotic behavior, r " Opr 1 q, A, E, u, φ " Op1q, B " Opr 1´1 q. (6.44) We can now trade the unknown r in the last of (6.40) for χ " re´E R satisfying Note that the vanishing of the Opr 1´1 q terms follows from non trivial cancellations. Except for the equation for r, which we have just discussed, the RHS of (6.40) and (6.43) are all Opr 1´2 q. We thus have where A 0 , B 0 , E 0 , u 0 , x A 0 , r 1 , χ 1 are all functions of u 1 , x 1A . These fall-offs allow us to write for µ " u, A, and where g 0 pu 1 , x 1A q " gpu 1 , 8, x 1A q, x µ 0 " x µ pu 1 , 8, x 1A q. Even though it will not be explicitly needed in the sequel, equations (6.40) and (6.43) can be used to work out the next to leading orders, At this stage, the unknowns are fixed up to A 0 , B 0 , E 0 , , R 1 pχ 1 q, u 0 , x A 0 as functions of We now have to require l 1 µ " δ u 1 µ . After having imposed l 1 " B Br 1 , one has in particular that l 1 r 1 " 0. This follows from l 1 Br 1 on the one hand and from Bx ν on the other. For the remaining components of l 1 µ it is enough to verify that l 1 u 1 " 1ò pr 1 0 q, l 1 A " opr 1 0 q since solutions are transformed into solutions under local Lorentz and coordinate transformations. In particular, de 1 a`Γ 1 a b e 1 b " 0, and for a " 2, de 1 11ν 1 " 0. This reduces to B r 1 l 1 ν 1 " B ν 1 l 1 r 1 , and thus to B r 1 l 1 u 1 " 0 " B r 1 l 1 A . Extracting the leading order from together with the complex conjugate of the last equation. When using that the change of coordinates needs to be invertible at infinity, these relations are equivalent to e E R0 " together with the complex conjugate of the last relation.
We now need the transformation laws of τ , σ and ρ, which are obtained from the matrix components 21 of the last equation of (6.17) for a " 2, 3, 4. This gives τ 1 " aapd 2 τ`c 2 ν´2cdγq´bapd 2 σ`c 2 µ´2cdβq´abpd 2 ρ`c 2 λ´2cdαqd ∆ 1 pcq´c∆ 1 pdq, (6.50) σ 1 "´capd 2 τ`c 2 ν´2cdγq`dapd 2 σ`c 2 µ´2cdβq`cbpd 2 ρ`c 2 λ´2cdαqd δ 1 pcq´cδ 1 pdq, (6.51) In order to proceed we need the asymptotic behavior of ∆, δ 1 , δ 1 . Using e 1µ a " Λ a b e b ν Bx 1µ Bx ν for a " 2, 3, we get n 1µ " bbDpx 1µ q`aa∆px 1µ q´rbaδpx 1µ q`c.c.s, m 1µ "´dbDpx 1µ q´ca∆px 1µ q`daδpx 1µ q`cbδpx 1µ q. (6.53) Explicitly, this gives On-shell the new tetrads need to have the same form in the new coordinates than they had in the old. This implies in particular together with the complex conjugates of these equations. In addition the requirement that the leading part of the metric remains conformally flat implies In order to work out the term on the RHS of τ 1 in (6.50) of order Opr 1´1 q, one needs in particular n 1 r 1 above. Requiring this term to vanish gives When using (6.49), this coincides with the second of (6.55).
Requiring that the tems of order r 1´2 in ρ 1 in equation (6.52) vanish yields Finally, to leading order, the transformation law of σ in (6.51) yields In summary, we see that all the unknowns A 0 , A 0 , B 0 , B 0 , χ 1 pR 1 q, E R0 are determined by the change of coordinates at infinity and by E I0 . The Jacobian matrices are given bÿ Note that here and in the following, when considered as a function of pu, ζ, ζq, E 0 is explicitly given by E 0 pu 1 0 pu, ζ, ζq, ζ 1 0 pζq, ζ 1 0 pζqq. Note also that the right lower corner of (6.61) is equivalent to the transformation law of P , P 1 pu 1 , ζ 1 , ζ 1 q " P pu, ζ, ζqe´Ē 0 Bζ In particular, when putting all results together, the subleading term of the rescaled radial coordinate is given by For notational simplicity, we drop in the next sections the subscript 0 on the asymptotic change of coordinates and on the complex Weyl parameter.

Combined extended BMS4 group with complex rescalings
From the top left corner of the first matrix of (6.61), it follows that E R is determined by upu 1 , ζ 1 , ζ 1 q and, conversely, that the knowledge of such a function allows one to recover the complete change of coordinates, up to an arbitrary functionû 1 pζ 1 , ζ 1 q, Note that the point with coordinates pû 1 pζ 1 , ζ 1 q, ζ 1 , ζ 1 q corresponds to p0, ζ, ζq.

Action on solution space
Putting the results of the previous subsections together, the transformation law of the data characterizing asymptotic solution space is contained in ı .

(6.86)
In particular for instance, if P, P 1 do not depend on u, u 1 , the transformation law of the asymptotic shear under a pure supertranslation reduces to σ 1 0 " σ 0`ð 2 α, u 1 " u`α, ζ 1 " ζ, (6.87) while the transformation law of the news under a pure superrotation is with u 1 , E given in (6.80). If furthermore, we work with respect to the Riemann sphere, P F " P R " 1, this reduces to where the Schwarzian derivative is tF, xu " B 2 x lnpB x F q´1 2 pB x lnpB x F qq 2 . Let us now analyze in more details the evolution equations (6.33). We start with unit scaling factors, P " P " 1, so that in particular the leading part of the metric on a spacelike cut of I`is the one on the Riemann sphere, ds 2 "´2dζdζ. In this case, (6.33) and (6.34) reduce to and In a first stage, these equations may be trivially solved in terms of integration functions r Ψ 0 aRI " r Ψ 0 aRI pζ, ζq for a " 0, 1, 2 as follows: and and the expressions for Ψ 0 1R in terms of r Ψ 0 1RI , r Ψ 0 2RI , σ 0 R , 9 σ 0 R and for Ψ 0 0R in terms of r Ψ 0 0RI , r Ψ 0 1RI , r Ψ 0 2RI , σ 0 R , 9 σ 0 R can be worked out recursively. For later use, we introduce instead the integration functions Ψ 0 aRI " Ψ 0 aRI pζ, ζq for a " 0, 1, 2 defined by to the non reduced quantities amounts to applying the combined complex rescaling to the reduced ones. In other words, only P, P , r u change while σ 0 R , 9 σ 0 R , Ψ 0 aR are unchanged as a function of their variables, while Ψ 0 iRI are completely unchanged. More generally, the transformation law of σ 0 R pr uq, 9 σ 0 R pr uq, Ψ 0 aR pr uq, a " 0, . . . , 4 under the extended BMS group combined with complex rescalings simplifies to σ 10 R " p where the primed quantities depend on r u 1 " J´1 2 pr u`βq, ζ 1 , ζ 1 , while the unprimed ones depend on r u, ζ,ζ. These transformations simplify for the standard BMS group since the Schwarzian derivative vanishes for this case.
For the transformation law of r Ψ 0 iRI , we find r Ψ 10 2RI " p The transformation laws of Ψ 0 2RI , Ψ 0 1RI can be obtained from that of r Ψ 0 2RI , r Ψ 0 1RI by using the first two relations of (6.99) and equation (B.2), respectively (B.3) of Appendix B.

Discussion
In this work, we have generalized finite BMS 4 transformations to include general holomorphic and antiholomorphic transformations as well as time-dependent complex rescalings. A further interesting generalization would be to abandon the reality conditions and consider the transformations discussed in this work in the context of H-space [44][45][46]. The approach we have followed here is systematic and straightforward but explicit computations are rather tedious and can presumably be simplified in a more suitable setup. Extracting physical consequences from these transformation laws should be much more rewarding. We conclude with some comments on why this should be the case.
The residual symmetry group we have investigated acts on the general asymptotically flat solution space in the sense of Newman-Unti [27], containing not only the Kerr black hole [47] but also Robinson-Trautman waves [48,49]. In this context, the analog of the time coordinate r u used here has been introduced previously in [50] in order to express the latter solutions in terms of a Bondi coordinate system where the conformal factor is the one for the unit sphere.
The transformations also naturally act on the would-be conserved BMS currents including Bondi mass and angular momentum aspects, which are built out of the data considered here. In order to cover the most general case, the expressions considered for instance in [10] have first to be generalized to the case of a variable, complex, u-dependent factor P . This will be done in [43].
The relevance of the transformation formulas to the gravitational memory effect [51,52] as described in [53] (see [54,55] for recent discussions) is obvious. The question of what part of this effect is controlled by BMS transformations boils down to a question about suitable orbits of the BMS group. These problems will be discussed in more details elsewhere, together with other applications involving topology-changing mappings.