Finite BMS transformations

Journal of High Energy Physics, Mar 2016

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.

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Finite BMS transformations

HJE BMS transformations Glenn Barnich 0 1 2 3 4 Cedric Troessaert 0 1 2 3 4 Physique Theorique et Mathematique 0 1 2 3 4 0 Arturo Prat 514 , Valdivia , Chile 1 Campus Plaine C. P. 231, B-1050 Bruxelles , Belgium 2 Universite Libre de Bruxelles and International Solvay Institutes 3 which reduces to @ 4 nally a change of coordinates @ The action of nite BMS and Weyl transformations on the gravitational data at null in nity is worked out in three and four dimensions in the case of an arbitrary conformal factor for the boundary metric induced on Scri. Classical Theories of Gravity; Gauge Symmetry; Gauge-gravity correspon- - Finite 1 Introduction 2 3 Adapted Cartan formulation 5 3d asymptotically at spacetimes at null in nity 6 4d asymptotically at spacetimes at null in nity 4 3d asymptotically AdS spacetimes at spatial in nity Additional transformation laws in 4d 1 Introduction There are two main applications of two dimensional conformal invariance [1]. The rst consists in using Ward identities associated to in nitesimal symmetry transformations in order to constrain correlation functions. In the second application, starting from known quantities in a given domain, the nite transformations are used to generate the correthe globally well-de ned symmetry group [3{5] in terms of locally de ned in nitesimal transformations has been proposed and studied in [6{10]. In particular, their relevance for gravitational scattering has been conjectured. Physical implications in terms of Ward identities for soft gravitons have subsequently been developed in [11{14]. The aim of the present paper is to derive the nite 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 S2 R 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{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 nite extended BMS4 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 de nition of asymptotically at spacetimes in terms of conformal compacti cations [18, 19]. As a warm-up, we start by re-deriving the known nite transformations in three dimensions in the asymptotically anti-de Sitter and at cases. In the former case, one recovers the Schwarzian derivative as an application of the AdS3/CFT2 correspondence [20, 21]. In the latter case, one obtains the nite transformation laws for the Bondi mass and angular momentum aspects that have been previously obtained by directly integrating the in nitesimal 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 nite Penrose-Brown-Henneaux transformations in the terminology of [23, 24]. Explicit computations are done in the framework of the Newman-Penrose formalism [25, 26], as applied to asymptotically at four dimensional spacetimes at null in nity in [27, 28]. Standard reviews are [29{33]. To summarize the results for the simplest case when computations are done with the extended BMS4 group consists of superrotations = ( 0); respect to the Riemann sphere, i.e., when the metric on I + is taken as ds2 = 0due2 2d d , = ( 0) together with supertranslations u0 = 2 [u + ( ; )]. In particular, the asymptotic part of the shear, the news, and the Bondi mass aspect transform as e f 0; g ; 1 2 + 2 ) + 1 4 f 0; gf 0; g(ue + ) ; (1.3) where f ; g denotes the Schwarzian derivative. { 2 { (1.1) (1.2) In the Cartan formulation of general relativity, the fundamental elds are on the one hand, a vielbein, ea , together with its inverse ea and associated metric g = ea abeb , where ab is constant and, on the other hand, a Lorentz connection satisfying the metricity condition ra bc = 0, abc = ad d bc = their inverses. The associated connection 1-form is torsion and curvature 2-forms are given by T a = dea + a Local Lorentz transformations are described by matrices ab = abce c with ec = ec dx . The b ^ eb, Rab = d ab + a c ^ ab(x) with a c bc = b a. Un[ab]c. Indices are lowered and raised with ab and g der combined frame and coordinate transformations, referred to as gauge transformations below, the basic variables transform as and cb. (2.1) (2.2) (2.3) (3.1) (3.2) e0a (x0) = 0abc(x0) = where the last expression is equivalent to the transformation law for the connection 1-form, 0ab = a c c d bd + Equations of motion deriving from the variational principle S[e; ] = 1 with the metricity condition, the former implies 1 2 abc = (Dbac + Dcab Dabc); where the structure functions are de ned by Dcabec = (ea(eb ) T a = 0 is equivalent to Dcab = 00 0 1 0 0 and the triad ea = (l; n; m) with associated directional covariant derivatives denoted by (D; ; ). Note that this choice of ab implies di erent conventions and normalizations than used in previous works. In particular, g (see also e.g. [34] for slightly di erent conventions). It follows that Dl = l Dn = Dm = l 2 m; n + 2 m; n; l = l n = m = l 2 m; n + 2 m; n; l = l n = m = l 2 m; n + 2 m; n: In order to describe Lorentz transformations, one associates to a real vector v = vaea 2 symmetric matrix v^ = vabja, where bja are chosen as In this case, the spin connection can be dualized, !c = 14 ab 123 = 1 and abc = ad be cf def . The 9 real spin coe cients are de ned by = abc!c with r D = = = marla 2 2 2ac 2bd ad + bc 0 d 2 2 bd c a 2 2 2cd 1 2ab CA ; ac ad + bc = ab cd g = 1 C ; A ! 2 abvavb; bja bjb = 1 2 abcbjc); bjabjbbja = 1 2 bjb ; For g 2 SL(2; R), one considers the transformation gbjagT va = bja abvb; g T = g 1: Alternatively, one can use vq = v^ in order to describe real vectors by traceless 2 2 qj1 = tr vq2 = abvavb; qjaqjb = ( ab + abcqjc); qjaqjbqja = gqjag 1va = qja abvb; q In terms of !^ = ^ja!a, this is equivalent to Explicitly, for the spin coe cients encoded in { 5 { 3d asymptotically AdS spacetimes at spatial in nity Fe erman-Graham solution space In the AdS3 case, = 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 Fe erman-Graham gauge xing is to assume that = = = 0: (4.1) which is equivalent to 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 where a = (+; ). The associated cotriad is = 0. In order to compare with the general solution given in [7], one introduces an alternative radial coordinate r = e p2L , in terms of which m = p e3 = p L 2 dr: r Under these assumptions, the Newman-Penrose eld 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 solution is given by = k = r4 = p 1 2Lk (r4 2C1r2 + C12 C12 + C2C3); = C2C3: p 2C2r2 Lk ; 1 2L2 ; = = i p p 2C3r2 Lk ; . The general an a nely parametrized spatial geodesic. In this case, r[amb] = n[alb]( + ) 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 = (x+; x ; ), = 1; 2; 3 such that m is normal to the surfaces = cte and the coordinate is the suitably normalized a ne parameter on the geodesic generated by m, the triad takes the form The last two radial equations involving the spin coe cients, equations (A.3) and (A.8), simplify to and are solved through = + ; = ; = C4 r 3 k C1r + C5 k C2r ; = C5 r 3 k C1r + C4 k C3r : The last radial equations are (A.11) and (A.12). Their r-component are trivially satis ed while their components along x are of the same form than (4.8), l = l + n ; n = n l ; (4.10) m = n = n HJEP03(216)7 (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) K1+ K2+ ! K1 K2 ; which then implies that equation (A.4) reduces to while the components along x of equation (A.10) become K1a@aC5 4 L2 C1 = 0; K1a@aK2 Using the radial form of the various quantities, equations (A.2) and (A.6) are equivalent to HJEP03(216)7 which leads to matrix formed by these functions, Note that asymptotic invertibility of the triad is controlled by the invertibility of the (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) Because of invertibility of the matrix (4.12), equations (4.15) and (4.14) can be used to express C4; C5 and C1 in terms of K1a and K2a. The two equations in (4.13) then become dynamical equations for C2 and C3. Since we now have treated all Newman-Penrose equations, the solution space is parametrized by K1a; K2a and by initial conditions for C2 and C3. In the limit r going to in nity, the triad elements l and n given in (4.11) take the form l = r 1K1 + O(r 3), n = r 1K2 + O(r 3). With a change of coordinates on the cylinder, we can make the associated asymptotic metric explicitly conformally at. This amounts to the choice K1+ = 0; K1 = p2e '; K2+ = p2e '; K2 = 0: Introducing this into equations (4.14) and (4.15), we get C4 = p2e ' while the dynamical equations (4.13) reduce to With the extra conditions (4.16), the space of solutions is parametrised by three functions '; C2 and C3 de ned 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 C2 and C3 in a di erent way using the action of the asymptotic symmetry group below. { 7 { The residual gauge transformations are the nite 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 Using the a = 3 component of the transformation law of the triad, r0 the requirement m0 = p2L r0 is equivalent to Expanding for each coordinate, we get p r0 = = p2r p2r k k = (1 + 2AB) p2L = rst rewrite the last equation of (3.19) as In order to implement the gauge xing condition on the new spin coe cients !q30 = 0, we When a = 3 this becomes and is equivalent to three conditions on the Lorentz parameters, When suitably combining these equations, one nds = A(1 + AB)e2E + B ; = A(1 + AB) Be 2E + A3(1 + AB)e2E + A2B ; (4.26) = A(1 + AB)eE Be E 2A2(1 + AB)e2E 2AB : { 8 { (4.19) (4.20) (4.21) (4.22) (4.23) (4.24) (4.25) The set of equations (4.22) and (4.26) forms a system of di erential 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, Inserting this into the equations, we easily get A0B0er0 r0 1 + O(r0 3); where we have assumed rr0 > 0 asymptotically. At this stage, we have xed the radial dependence of all the unknown functions and we are left with six functions A0; B0; E0, r0; x0 of x0 . we already have m0r0 = We now have to require m0 = 3cec on the other. For the remaining components of m0 it is enough to verify pr20L . This follows from m0r0 = that m0 0 = o(r00) since solutions are transformed into solutions under local Lorentz and coordinate transformations. Indeed, de0a + r02 p2L r0 . Extracting the leading order from 0abe0b = 0, and for a = 3, de03 + 03be0b = 0. 033 0 = 0, (4.28) HJEP03(216)7 (4.29) (4.30) (4.31) (4.32) we then get The last condition we have to require is the asymptotically conformally at form of the new triad. This can be done by imposing r0e'0 p 2 The leading terms of e01 = Combining with equation (4.30), allows one to extract A0; B0; E0 in terms of the other E0 = ln 1 2 e ' r0 ; : (4.33) { 9 { It thus follows that the residual gauge symmetries are determined (i) by the change of to be orientation preserving @+0x0+ > 0 < @0 x0 , and (ii) by r0(x0+; x0 ). variables x = x0 (x0 ) at in nity, each depending on a single variable, which we assume For notational simplicity, we drop the subscript 0 on the change of variables at in nity and on the Weyl parameter in the next section. The group obtained in the previous section is the combined conformal and Weyl group and is parametrized by x0+(x+); x0 (x ); r(x0+; x0 ) : (4.34) The last equation of (4.32) encodes the transformation law of ', a rst, a second successive and the combined transformation respectively, the composition law is rc(x00+; x00 ) = rs(x00+; x00 ) + r(x0+; x0 ): This group reduces to the conformal group for xed conformal factor of the boundary metric: when ' = '0 it follows from (4.35) that r is determined by the change of variables at in nity, r = remains with the additive group of Weyl rescalings that amount here to arbitrary shifts of '. As discussed in section 4.1, the on-shell metric, triads and spin connections are entirely determined by the arbitrary conformal factor '(x+; x ) and the integration functions C2(x+; x ); C3(x+; x ) satisfying (4.18). To obtain the action of the group on the latter, we can extract the subleading terms of l0(x+) = 1beb+ and n0(x ) = 2beb . This gives which can also be written in terms of ' using equation (4.35). Note that, by construction, the transformed C20(x0+; x0 ); C3(x0+; x0 ) have to satisfy the transformed equations, i.e., equations (4.18) where all quantities, C2; C3; ', x ; @ are primed. In the particular case where ' = 0, equations (4.18) reduce to @+C2R = 0, @ C3R = 0 so that C2R = (8 GL)T (x ) and C3R = (8 GL)T++(x+). Applying the particular Weyl transformation x0 = x , r = '0, and removing all primes, we obtain from (4.37) that the general solution to the dynamical equations (4.18) for arbitrary ' is given by (4.35) (4.36) (4.37) (4.38) Solution space can thus also be parametrized by the conformal factor ' and the two integration functions T (x ) depending on a single variable each. The action of the asymptotic symmetry group on the latter can be extracted from equations (4.37), (x0 ) c (x ) + 24 fx0 ; x g ; c = 3L 2G ; in terms of the Schwarzian derivative for a function F of x, and with the characteristic values of the central charges for asymptotically AdS3 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. c 1 2 = = = 0: = ; 5 5.1 Solution space 3d asymptotically at spacetimes at null in nity The rst gauge xing conditions that we will assume are = = C3 1 1 2 r + C1 ; 2C22 ; = This is equivalent to 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 a nely parametrized null geodesic. In this case, r[alb] = always hypersurface orthornormal. It is a gradient if and only if ), so that l is a condition which will also be imposed in the following. Introducing Bondi coordinates x = (u; r; ); = 0; 1; 2 such that the surfaces u = cte are null with normal vector l, l = 0 and such that r is the suitably normalized a ne parameter on the null geodesics generated by l, the triad takes the form l = m = U + T The associated cotriad is e1 = ( W + T 1U V )du + dr T 1U d ; e2 = du e3 = T 1V du + T 1 d : (5.4) Under these assumptions, the Newman-Penrose equations (A.1){(A.6) x the r dependence of all spin coe cients according to (4.39) (4.40) The rst condition can be satsi ed by changing the a ne parameter r ! r + C1. We can then do a Lorentz transformation with a = d = 1, c = 0 and b = C2 in order to impose C2 = 0, and Note however that both of these last two transformations are only valid asymptotically. Requiring them to preserve the gauge xing 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 C1 = C2 = K2 = 0: Rede ning K1 = e ', the remaining equations, i.e., (A.7){(A.9) and (A.12), are equivalent to where C4 = K4; (5.6) (5.7) (5.8) (5.9) (5.10) (5.11) for Ci = Ci(u; ). When used in equations (A.10) and (A.11), the r dependence of the triad is T = W = K1 ; In order to solve the remaining equations, we will assume in addition that = + O(r 3); = O(r 2); V = O(r 1): HJEP03(216)7 These equations can be integrated directly, but we will again generate the solution by using the asymptotic symmetry group below. In this case, (5.5) and (5.6) simplify to = T = 1 2r e ' r ; ; = = 0; W = U = K3 r ; = K4 2r ; V = 0: 5.2 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 u = u(u0; r0; 0); r = r(u0; r0; 0); = (u0; r0; 0); (5.12) combined with an arbitrary local Lorentz transformation. The unknowns are A; B; E; u; r; Using the a = 1 component of the transformation law for the triad it follows that imposing l0 = r0 is equivalent to the radial equations, = 2B(1 + AB)T; = (1 + AB)2eE + B2e EW + 2B(1 + AB)U: of the last equation of (3.19) can be rewritten as The gauge xing on the new spin coe cients takes the form !q10 = 0. The component a = 1 1b!qb: This is equivalent to three conditions on the rotation parameters, which can be suitably combined to yield = 2B(1 + AB)eE ; = = B2e 2E B2e E 2B(1 + AB)e E + 2A2B(1 + AB)eE ; (5.16) 4AB(1 + AB)eE : The set of equations (5.14) and (5.16) forms a system of di erential 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 = re E = O(r0) satisfying 2B = 1 A2B2 + B2e 2EK4 + (1 + AB)e 2EK3 = 1 + O(r0 2): The solution is given by which, when introduced into the other radial equations, gives = r0 + 0(u; ) + O(r0 1); B = B0(u; )r0 1 + (A0B02 B0 0)r0 2 + O(r0 3); 2A0B0)r0 1 + O(r0 2); u = u0(u; ) = 0(u; ) B02e E0 r0 1 + O(r0 2); 2B0e ' E0 r0 1 + O(r0 2): (5.13) (5.14) At this stage, we have xed the radial dependence of all the unknowns and are left with We now have to require l0 = u0 . After having imposed l0 = @@r0 , one has in particular that lr00 = 0. This follows from the combination of lr00 = the remaining components of l0 it is enough to verify that lu00 = 1 + o(r00), l0 = o(r00) since 1cec . For the equation of motion de0a + 0abe0b = 0 for a = 2 implies @r0 l 0 leading order from e03 = ( 2acW + 2bd) + 2ac + ( 2acU + ad + bc) 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 e0u3 and e03 yield + ( c2U + cd) e E0 = we get We still have to impose three conditions: V 0 = O(r0 1), 0 = The rst one is a condition on the triad and can be imposed by requiring e0u3 = O(1). More 21r0 + O(r0 3), 0 = O(r0 2). generally, we have Note in particular that our choice of parametrization for the Lorentz rotations leads to @@ 00 > 0. The rst equation is equivalent to V 0 = O(r0 1) while the second one gives the transformation law of '. Combining (5.23) with (5.21), we obtain To implement the last two conditions, we will use the transformation law of !3 given in the last equation of (3.19). Imposing (!q30)11 = O(r0 2) and (!q30)21 = 21r0 + O(r0 3), we get q A0 + 1 0E + B 0A = O(r0 2); (5.20) (5.21) (5.22) (5.23) (5.24) (5.25) (5.26) B0e 2 B02e E0 From the general solution, we have 0 = U 0@r0 + T 0@0 where U 0 = O(1) and T 0 = e '0 r0 1 + O(r0 2). Inserting this into the two equations we can extract the value of A0 and 0 : 2r0 1 r02 1 2 0 + 2A0B0 B(1 + AB) 0E + 0B = O(r0 3): The asymptotic symmetry group is thus parametrised by three functions u0(u0; 0), 0(u0; 0) and E0(u0; 0) satisfying the constraints Note that, when taking these into account, the Jacobian matrices for the change of coordinates at in nity are ! ; For notational simplicity, we will drop the subscript 0 on the functions determining the change of coordinates at in nity and on the Weyl parameter in the next two section. From equation (5.27), it follows that E(u0; 0) is determined by the function u(u0; 0) and, conversely, that the knowledge of such a function E allows one to recover the complete change of coordinates, up to an arbitrary function u^0( 0), u(u0; 0) = dv0 e E : Z u0 u^0 Z u u^ Note that the point with coordinates (u^0( 0); 0) in the new coordinate system is described by (0; ) in the original coordinate system. When considering the inverse transformation, we can write, u0(u; ) = dv eE ; where E0 is now considered as a function of the original coordinate system through E0(u0(u; ); 0(u; )) and the point with coordinates (u^( ); ) is described by (0; 0) in the new coordinate system. Equation (5.24) is equivalent to the transformation law of eld ', '0(u0; 0) = '(u; ) + E(u0; 0) + ln This can be used to trade u^( ) for which can be inverted since the integrand is positive. The combined BMS3 and Weyl group can be parametrized by Note that the transformation law of Z 0 u^ ( ) = dv e '(v; ); ( 00( ); ( ); E(u0; 0)): e u(u; ) = dv e '(v; ); Z u 0 (5.27) (5.28) In particular, if ( ); E(u0; 0); s( 0); Es(u00; 00); c( ); Ec(u00; 00); are associated to a rst, a second successive and their combined transformation respectively, equations (5.31) and (5.34) imply that e u(u; ) + ( ) : (5.35) c( ) = s( 0) + ( ); and parametrize the combined BMS3 and Weyl group by In this case, equation (5.35) and the rst of equation (5.36) are replaced by For xed di eomorphism on the circle, 0 = , the rst of (5.36) describes the abelian subgroup of supertranslations, while, if in addition one restricts to the subgroup without supertranslations, i.e., when all 's vanish, so do the u^'s and u is unchanged. The second of (5.36) then describes the abelian subgroup of Weyl rescalings. Alternatively, one can de ne U (u; ) = e'ue; (u; ) = e' ; ( 0( ); (u; ); E(u0; 0)): U 0(u0; 0) = eE(u0; 0) U (u; ) + (u; ) ; c(u; ) = e E(u0; 0) s(u0; 0) + (u; ): Note that if ' does not depend on u then Ue = e 'u, = e 'u^( ) whereas U = u and ( ) = u^. If furthermore '0 does not depend on u0, then neither does E and u0(u; ) = eE( 0)(u + ). The standard de nition of the BMS3 group is then recovered when the conformal factor is xed to be zero, i.e., when '(u; ) = 0 = '0(u0; 0), in which case it follows from equation (5.31) that the Weyl transformations are frozen to eEF = @@ 0 . 5.4 Action on solution space Solution space is parametrized by the three functions ', K3, K4 satisfying the evolution equations (5.10). The action of the group on the conformal factor ' has already been computed in the previous section. We can extract the transformation law of K4 from (!q30)12 and the one of K3 from the second order of e01, K40 = e 2EK4 + A02 + 2e '0 @ 0 A0 + 2B0e 2E ' K30 = e 2EK3 + 2e 2EB0K4 (5.37) (5.38) (5.39) (5.40) By construction, the transformed quantities have to satisfy the transformed equations, i.e., equations (5.10) where all quantities, K3; K4; ', u; , @u; @ are primed. In the particular case where ' = 0, equations (5.10) reduce to @uK4R = 0, @uK3R = tion u = R0u0 dv0e '0(v0; 0), we obtain from (5.40) that (5.41) 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 ' is given by K3 = (16 G)e 2' j( ) + u@ p( ) + 2@ uep( ) e 1 The nal parametrisation of the solution space studied in section 5.1 is given by the conformal factor ' and the two functions p( ) and j( ). Their transformation laws under the combined BMS3 and Weyl group is given by p0( 0) = j0( 0) = The central charges have the characteristic values for asymptotically at 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 K3; K4 amounts to applying the combined Weyl transformation to p; j with the associated change of ue. Their transformations under the BMS3 group agree with those derived by di erent methods in [22, 38]. 00 1 0 ab = BBB10 00 00 0 1 0 C C : 1C A 0 0 (5.42) (5.43) (6.1) 6 6.1 4d asymptotically at spacetimes at null in nity Newman-Penrose formalism in 4d In four dimensions with signature (+; ; ; ), we use The di erent elements of the null tetrad are denoted by ea = (l; n; m; m), with the associated directional covariant derivatives denoted by (D; ; ; ). In particular, g The 24 independent abc's are parametrized through 12 complex scalars, marla 12 (narla marma) marna = = = = 311 312 313 314 where the associated complex conjugates are obtained by exchanging the indices 3 and 4. In order to describe Lorentz transformations in four dimensions in terms of a null tetrad, one associates to a real vector v = vaea, with v1; v2 hermitian matrix v = vabja, where the bja are chosen as 2 R; v4 = v3 2 C, a 2 2 2 abvavb; bjbT bja + bjaT bjb = ab ; bjb bjaT + bja bjbT = ab ; bja(bjb + bjbT )bjaT = 0; (6.5) In this case where 1 r D b bj1 = = g = (6.3) (6.4) (6.6) (6.7) (6.8) (6.9) For an element g 2 SL(2; C), one considers the transformation gbjagyva = bja abvb; g T = g 1 : More explicitly, if ab = BBB@ accc dbdd acdd dbccCCA C ; ca db cb da 0 dd ab = BBB bdbb ca da ac bc dc ba cd1 abC cb CA ad C ; where the rst index is the lign index. The standard three classes of rotations [30] are then given by class I for which l0 = l; m0 = m + Al; m0 = m + Al; n0 = n + Am + Am + AAl: a = 1 = d, c = 0, b = a = 1 = d, b = 0, c = A, A 2 C, B, B 2 C, class II for which n0 = n; m0 = m + Bn; m0 = m + Bn; l0 = l + Bm + Bm + BBn: the transformation law of Lorentz connection becomes More explicitly, for the spin coe cients encoded in g = e E=2 b g dgT : ! ; !2 = b ! ; !3 = b ! ; !4 = b !ba0 = acg!cg T b Finally, the SL(2; C) group element corresponding to a combined rotation II I III Alternatively, one can use vq = vb in order to describe real vectors. The associated qj1 = one nds basis is so that where b !1 = 1 q !1 = ! ; ! : (6.10) (6.11) 2 abvavb; qjbT qja + qjaT qjb = ab ; qjb qjaT + qja qjbT = ab ; qja(qjb + qjbT )qjaT = 0: (6.16) In this case, we have gqjag 1va = qja abvb; q For the Weyl scalars, we follow the conventions of [33, 39], which di er by a sign from those of [30] and those of [25, 27] (when taking into account in addition the correction for 2 given in [26, 29]). If Cabcd denote the components of the Weyl tensor and ABCD the associated Weyl spinor, 0 = C1313 $ 3 = C1242 $ 0000; 0111; 1 = C1213 $ 4 = C2424 $ 0001; 1111: 2 = C1342 $ 0011; Their transformations law under Lorentz rotations can be either worked out directly by using Ca01a2a3a4 = a1 b1 : : : a4 b4 Cb1b2b3b4 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) as in (6.9) corresponds to 0A = gAB B with When taking into account the complete symmetry of the Weyl spinor, one gets directly from 0A1A2A3A4 = gA1 B1 : : : gA4 B4 B1B2B3B4 that 04 = a4 ; ; ; ; Newman-Unti solution space The gauge xing conditions at null in nity1 that are usually assumed correspond to imposing the six real conditions encoded in = = = 0. This is equivalent to requiring ab1 = 0 and can be achieved by a suitable Lorentz rotation. According to the de nition 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 a nely parametrized null geodesics. One then requires in addition that l is hypersurface orthonormal and a gradient, which yields 3 more conditions, = and = + , see, e.g., section 1.9 of [30]. This allows one to choose Bondi coordinates x = (u; r; xA), = 0; : : : ; 3, A = 2; 3, xA = ( ; ) such that the surfaces u = cte are null with normal vector l, l = 0 and that r is the suitably normalized a ne parameter on the null geodesics generated by l. The tetrad then takes the form l = n = + U ; (6.22) which implies that g 0 = 1 ; g11 = 2(U 1We restrict the discussion to I +. !!); g1A = XA (! A + ! A ); gAB = ( A B + B A ): (6.23) On a space-like cut of I +, we use coordinates ; , and the metric e3 = XA Adu AdxA; e4 = XA Adu AdxA: ds2 = ABdxAdxB = 2(P P ) 1d d ; e1 = e2 = du; [U + XA(! A + ! A)]du + dr + (! A + ! A)dxA; with P P > 0. For the unit sphere, we have coordinates and = cot 2 e i in terms of standard spherical The covariant derivative on the 2 surface is then encoded in the operator PS( ; ) = p (1 + ): where g; g 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", gP = 0 and that [g; g] s = R s ; According to [25{27], once the conditions = = 0 are xed and coordinates u; r; ; such that l = u, l = r are chosen, which implies in particular also that = 0 = , the leading part of the asymptotic behaviour given in (6.30) follows from the equations of motion, the condition 0 = 0 0 r 5 + O(r 6) and uniform smoothness, i.e., a standard restriction on the functional space imposing how the fall-o conditions in r behave with respect to di erentiation. In addition, the choice of a suitable radial coordinate is used to put to zero the term in of order r 2, while by a choice of coordinates xA, the leading part r2 of the spatial metric is set to be conformally at, and the constant part of XA to vanish. Finally, the leading order r 1 of 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. 1 2 s 2 (6.24) (6.25) (6.26) (6.27) (6.28) (6.29) formalism and that x4 ponents of the Weyl tensor are given by 0r 5 + O(r 6); 0 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 g operator is taken to agree with the de nition used in [33]. Furthermore, = x3 + ix4 and r = 2@. 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 The asymptotic expansion of on-shell spin coe cients, tetrads and the associated com1 2 1 6 g 00r 5 + O(r 6); g 10r 4 + O(r 5); g 20r 3 + O(r 4); g 30r 2 + O(r 3); 0 0r 3 + O(r 5); 0r 2 + 0r 3 + O(r 4); 1 = P 0r 2 + O(r 4); = P (r 1 + 0 0r 3) + O(r 4); ! = g 0r 1 0 g 0 + 12 01 r 2 + O(r 3); h h 1=2 1=2 e 1=2 1=2 e 3=2 1=2 e 1 1 e 1=2 3=2 e 2 0 5=2 1=2 3=2 3=2 1=2 5=2 and 0 = 0 = 1 2 0) 0; 2 0 g 1 2 1 2 gg ln(P P ) = R 4 ; 0 = 1 0 =g( 0 + 0); 04 =g 0 (6.33) (6.35) In this approach to the characteristic initial value problem, freely speci able initial data at xed u0 is given by 0(u0; r; ; ) in the bulk (with the assumed asymptotics given above) and by ( 02 + 0 2)(u0; ; ), 01(u0; ; ) at I +. The asymptotic part of the shear 0(u; ; ) 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 eld s;w of spin and conformal weights (s; w), one can associate a eld eh;h of conformal dimensions (h; h) through 1 2 s;w = P hP h h;h; (h; h) = (s + w); (s w) : (6.36) The conformal dimensions of the various quantities used here are given in table 2. When expressed in these quantities, (6.34) and (6.35) become 0 e 2 e e 0 = 0 = 0 e 2 = e 1 2 2 1 2 e 0 e 0 + e e e 0 = e 0 1 2 1 2 and 3 2 e e 2; e e 3; 0 0 e e 4; Below, during the construction of the solution to the evolution equations (6.34) and (6.35), we will construct improved elds of this type that take due care both of the additional u-dependence 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 coe cients 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 u = u(u0; r0; x0A); r = r(u0; r0; x0A); x A = xA(u0; r0; x0A); and the unknowns are A; A; B; B; E; E; u; r; xA as functions of u0; r0; x0A. Using the a = 1 component of the transformation law abeb ; it follows that imposing l0 = (dcm + c:c:), or more explicitly, r0 is equivalent to replacing the l.h.s. by @@xr0 . This gives = BBe ER ; = BBe ER XA + B(1 + AB)eiEI A + c:c: ; = j1 + ABj2eER + BBe ER U + B(1 + AB)eiEI ! + c:c: : In order to implement the gauge xing conditions in the new coordinate system, or equivalently !q10 = 0, we rewrite the last of (6.17) as and require, for a = 1, 1b!qb: ab!qb = (6.37) (6.38) (6.39) (6.40) (6.41) (6.42) (6.43) HJEP03(216)7 More explicitly, the conditions on the rotation parameters are dD0a bD0c = 1b(!qb)11; dD0b bD0d = 1b(!qb)12; aD0c cD0a = 1b(!qb)21; where 1b(!qb)11 = cc 1b(!qb)12 = cc 1b(!qb)21 = equations, one nds dc dc cd = BBe ER + B(1 + AB)eiEI + B(1 + AB)e iEI ; cd = BBe ER + B(1 + AB)eiEI + B(1 + AB)e iEI ; cc + dc + cd = BBe ER B(1 + AB)eiEI B(1 + AB)e iEI : Note that the additional equation involving 1b(!qb)22 = q 1b(!b)11 on the r.h.s. follows from the rst equation when using ad bc = 1. When suitably combining these The set of equations (6.41) and (6.44) forms a system of di erential 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, We can now trade the unknown r in the last of (6.41) for = re ER satisfying = j1 + ABj2 + BBe 2ER U + B(1 + AB)e E! AeE 1b(!qb)21) + c:c: = 1 + O(r0 2): (6.46) Note that the vanishing of the O(r0 1) terms follows from non trivial cancellations. Except for the equation for r, which we have just discussed, the r.h.s. of (6.41) and (6.44) are all O(r0 2). We thus have = E A e e E 1b(!qb)12; = 2AeE 1b(!qb)21 2 1b(!qb)11: (6.44) (6.45) (6.47) A = A0 + O(r0 1); u = u0 + O(r0 1); r0B = B0 + O(r0 1); x A = x0A + O(r0 1); r = eER0 r0 + r1 + O(r0 1) () = r0 + 1 + (r0 1); E = E0 + O(r0 1); = 1b!qb 1beb () g(u0; r0; x0A) = e Rr10 dr~( 1bg!qb)(u0;r~;x0A)g0(u0; x0A); () x (u0; r0; x0A) = x (u0; 1; x0A) dr~( 1beb )(u0; r~; x0A); Z 1 r0 for = u; A, and where g0(u0; x0A) = g(u0; 1; x0A), x0 = x (u0; 1; x0A). Even though it will not be explicitly needed in the sequel, equations (6.41) and (6.44) can be used to work out the next to leading orders, u = u0 = 0 B0B0e ER0 r0 1 + O(r0 2); B0e E0 P r0 1 + O(r0 2); We now have to require l0 = u0 . After having imposed l0 = @@r0 , one has in particular that lr00 = 0. This follows from lr00 = 1cec on the other. For the remaining components of l0 it is enough to verify that lu00 = 1 + o(r00), lA0 = o(r00) since solutions are transformed into solutions under local Lorentz and coordinate transformations. In particular, de0a + Contracting with e0 1 0abe0b = 0, and for a = 2, de01 + 01be0b = 0. 01b1e0b 0 011 0 = 0. This e02 cc[U + XA(! A + ! A)] + dd + cdXA + cc + cc(! A + ! A) cd A ; (6.48) we get c 0(d); c 0(d); together with the complex conjugate of the last equation. When using that the change of coordinates needs to be invertible at in nity, these relations are equivalent to eER0 = e E0 gu00; together with the complex conjugate of the last relation. We now need the transformation laws of , and , which are obtained from the matrix components 21 of the last equation of (6.17) for a = 2; 3; 4. This gives 0 = aa(d2 + c2 2cd ) ba(d2 + c2 2cd ) ab(d2 + c2 2cd ) 0 = ca(d2 + c2 2cd ) + da(d2 + c2 2cd ) + cb(d2 + c2 2cd ) (6.49) (6.50) (6.51) (6.52) 0 = 2cd ) + bc(d2 2cd ) + ad(d2 + c2 In order to proceed we need the asymptotic behavior of for a = 2; 3, we get ; 0; 0. Using e0a = n0 = bbD(x0 ) + aa (x0 ) [ba (x0 ) + c:c:]; m0 = dbD(x0 ) ca (x0 ) + da (x0 ) + cb (x0 ): Explicitly, this gives ln P P ER0 r0 + O(r00); n0A = X0A = e ER0 @x00A + O(r0 1); 1 2 (6.54) (6.55) (6.56) (6.57) (6.58) (6.59) HJEP03(216)7 ln(P P ) ER0 e E0 gER0 + O(r0 1); 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 P 0 = e E0 g 00; ln(P P ) ER0 gu00 + e E0 gER0; together with the complex conjugates of these equations. In addition the requirement that the leading part of the metric remains conformally at implies When used in (6.49) this leads to : In order to work out the term on the r.h.s. of 0 in (6.51) of order O(r0 1), one needs in particular n0r0 above. Requiring this term to vanish gives A0 = 1 P P When using (6.50), this coincides with the second of (6.56). Requiring that the tems of order r0 2 in 0 in equation (6.53) vanish yields 1 = B0B0e ER0 @u ln(P P ) + A0B0 + 2A0B0 + g0B0 2B0g0ER0: Finally, to leading order, the transformation law of in (6.52) yields 00 = e ER0+2iEI0 0 A0B0 g0B0: In summary, we see that all the unknowns A0; A0; B0; B0; 1(R1); ER0 are determined by the change of coordinates at in nity and by EI0. The Jacobian matrices are given by B B B P C ; C P 0 P C : C C C C A Note that here and in the following, when considered as a function of (u; ; ), E0 is explicitly given by E0(u00(u; ; ); 00( ); 00( )). Note also that the right lower corner of (6.62) is equivalent to the transformation law of P , P 0(u0; 0; 0) = P (u; ; )e E0 @ 00 in (6.25) and of g are given by (ds2)0 = e2ER0 (ds2); g0 s = e E0 g s gE0 ln P ) s : In particular, when putting all results together, the subleading term of the rescaled radial coordinate is given by ] e 2ER0 ggu00: (6.65) For notational simplicity, we drop in the next sections the subscript 0 on the asymptotic change of coordinates and on the complex Weyl parameter. (6.60) (6.61) (6.62) (6.63) (6.64) dv0 e ER : dv eER ; Z u0 u^0 Z u u^ Z 0 u^ ( ; ) = dv (P P ) 21 ; ( 0( ); 0( ); ( ; ); E(u0; 0; 0)); u(u; ; ) = Z u 0 dv(P P ) 21 (v; ; ); Note that the point with coordinates (u^0( 0; 0); 0; 0) corresponds to (0; ; ). After inverting, one can write, is given by (0; 0; 0) in the new coordinate system. For a eld P transforming as in (6.63), we trade u^( ; ) for which can be inverted since the integrand is positive. The extended BMS4 group combined with complex rescalings can be parametrized by since ( ; ) determines u^( ; ) and one then gets u0(u; ; ) from (6.67). De ning From the top left corner of the rst matrix of (6.62), it follows that ER is determined by u(u0; 0; 0) and, conversely, that the knowledge of such a function allows one to recover the complete change of coordinates, up to an arbitrary function u^0( 0; 0), (6.66) (6.67) (6.68) (6.69) (6.70) (6.71) (6.72) (6.73) (6.74) its transformation law is simply 1 ue0(u0; 0; 0) = J 2 ue(u; ; ) + ( ; ) ; J = Together with (6.63), this implies in particular that, if ( ; ); E(u0; 0; 0); and the same quantities with a superscript s and a superscript c are associated to a rst, a second successive and the combined transformation respectively, we have c( ; ) = J 21 s( 0; 0) + ( ; ); Ec(u00; 00; 00) = Es(u00; 00; 00) + E(u0; 0; 0): U (u; ; ) = (P P ) 21 u; e (u; ; ) = (P P ) 21 ; and parametrize the extended BMS4 combined with complex rescaling through Equation (6.71) and the rst of equations (6.73) are then replaced by ( 0( ); 0( ); (u; ; ); E(u0; 0; 0)): U 0(u0; 0; 0) = eER(u0; 0; 0) U (u; ; ) + (u; ; ) ; c(u; ; ) = e ER(u0; 0; 0) s(u0; 0; 0) + (u; ; ): = A + B + B + C 1 + A; C 2 R; B 2 C: U = u and ( ; ) = In the case when P does not depend on u, ue = (P P ) 21 u and (P P ) 21 u^, whereas u^. If furthermore P 0 does not depend on u0, then neither does E and u0(u; ; ) = eER( 0; 0)(u + ); e ER gu0 = g + gER(u + ): some xed function PF of its arguments, it follows from (6.63) that complex rescalings are frozen to E = : In this case, a pure supertranslation is characterized by while a pure superrotation is characterized by 0 = ; E = 1; u0(u; ; ) = u + ; e ER gu0 = g ; 0 = 0( ); eE = ; u0 = eER u; e ER gu0 = gERu: The standard de nition of the BMS4 group is then recovered when (i) standard Lorentz rotations are described through fractional linear transformations (see e.g. [41] for details), 0 = a + b c + d ad bc = 1; a; b; c; d 2 C; (ii) the conformal factor is xed to be that for the unit sphere, PF = PS, in which case eERS = 1 + (a + b)(a + b) + (c + d)(c + d) ; eiEIS = c + d c + d ; (iii) supertranslation are expanded in spherical harmonics, S = P l;m lmYlm( ; ) with ordinary translations corresponding to the terms with l = 0; 1 and are explicitly described by A pure complex rescaling is characterized by HJEP03(216)7 Z u 0 0 = ; u0 = dv eER ; P 0 = P e E : (6.75) (6.76) (6.77) (6.78) (6.79) (6.80) (6.81) (6.82) (6.83) (6.84) 00 = e ER+2iEI h 004 = e 3ER 2iEI 04 003 = e 3ER iEI h 0 3 e ER gu0 04i; Putting the results of the previous subsections together, the transformation law of the data characterizing asymptotic solution space is contained in 0 + g(e ER gu0) 0)(e ER gu0)i; (6.85) 0) g(e ER gu0) (e ER gu0)(@u + 0 0)(e ER gu0) i; (6.86) 000 = e 3ER+2iEI h 00 4e ER gu0 01 +6(e ER gu0)2 02 4(e ER gu0)3 03 +(e ER gu0)4 04i: In particular for instance, if P; P 0 do not depend on u; u0, the transformation law of the asymptotic shear under a pure supertranslation reduces to while the transformation law of the news under a pure superrotation is 00 = 0 + g2 ; u0 = u + ; 0 = ; 00 = e 2E [ 0 + (g2ER (gER)2]; with u0; E given in (6.81). If furthermore, we work with respect to the Riemann sphere, PF = PR = 1, this reduces to 00 = 0 + 1 2 u0 = J 21 u; 0 = 0( ); Let us now analyze in more details the evolution equations (6.34). We start with unit scaling factors, P = P = 1, so that in particular the leading part of the metric on a space-like cut of I + is the one on the Riemann sphere, ds2 = 2d d . In this case, (6.34) and (6.35) reduce to and 0 2R R0 = 0 = 0 2 0 R 0 3R = R0 = 0 R ; 2 0R + _ R R 0 0 0R = _ 0R; _ 0 0 R R ; 0 4R = 0R; @u 0R = @ 10R + 3 R0 02R; 0 R0 04R; (6.87) (6.88) (6.89) (6.90) (6.91) (6.92) In a rst stage, these equations may be trivially solved in terms of integration functions 0 0 e aRI = e aRI ( ; ) for a = 0; 1; 2 as follows: 0 0 2R = e 2RI + 0 3R + 0 0 R 4R ; 0 e 2RI 0 2 0R + _ R R 0 0 _ 0R R0)(0); 0 0 1R = e 1RI + 0 2R + 2 R0 03R ; HJEP03(216)7 0 0 0R = e 0RI + 0 1R + 3 R0 02R ; 0 1R 0R in terms of R can be worked out recursively. For later use, we introduce instead the integration functions 0 0 aRI ( ; ) for where and and the expressions for e 0RI ; e 1RI ; e 2RI ; 0 ; _ 0 0 0 R 0 0 0R = 2RI Z u 0 Z u 0 dv dv 1 2 Z v 0 Z u Z u 0 0 dv R0(3 dv 3 R0 Z v 0 dw _ R0 _ 0R; 2 0 2RI Z v 0 Z v 0 Z w 0 dw _ R0 _ 0R 1 2 dx _ R0 _ 0R: Z u 0 Z u 0 Z u 0 (6.93) (6.94) (6.95) (6.96) (6.97) (6.98) (6.99) (6.100) 0 2R = 0 1R = 0 2RI 2 0 R R0 _ 0R + dv _ R0 _ 0R; 0 2RI = 0 2RI ; 0 2RI 0 R Z u 0 1 2 R R 3 R0 _ 0R R 2 R R 1 2 R R We have 2 0R + R0 _ 0R](0); 0 0 1RI = e 1RI + 0 0 0RI = e 0RI : 1 2 ( 4 G)MR = 02R + R0 _ 0R + @2 0R = 0 2RI + dv _ R0 _ 0R: (6.101) Z u 0 Note that this expression contains the additional term @2 0R as compared to the more convential choice, see e.g., equation (4.18) of [10] with f = 1 (up to a global minus sign due to the change of signature). More details on this quantity, and more generally on the would-be conserved currents and their transformation laws, will be given elsewhere [43]. In order to generate the general solution to (6.34) and (6.35) for arbitrary scaling factors P; P from the one with P = 1 = P , we apply a pure complex rescaling, without superrotations nor supertranslations to the solution above, i.e., we take eE = P 0 1, u0 = R0u dv eER, 0 = . In this particular case, 0 = 0; e ERgu0 = As a consequence, one nds from the transformation laws that the general solution to (6.34) and (6.35) is given by HJEP03(216)7 0 = P 21 P 23 h R0(ue) 2 i 0 = P 2 _ 0R(ue) 1 2 04 = P 25 P 21 h 04R(ue)i; 03 = P P 2 i (6.103) where all functions depend on u; ; , except where explicitly indicated that the dependence on u is replaced by a dependence on ue(u; ; ). In particular, this means that R0(ue); _ 0R(u); 0aR(ue); a = 0; : : : ; 4 and 0RiI , i = 0; 1; 2, e are invariant under complex rescalings. Indeed, applying a complex rescaling to the non their variables, while reduced quantities amounts to applying the combined complex rescaling to the reduced ones. In other words, only P; P ; ue change while 0 ; _ 0R; 0aR are unchanged as a function of R 0 iRI are completely unchanged. More generally, the transformation law of R0(ue); _ 0R(ue); 0aR(ue); a = 0; : : : ; 4 under the extended BMS group combined with complex rescalings simpli es to 1 0 2 + f ; g(u~ + ) ; Schwarzian derivative vanishes for this case. where the primed quantities depend on ue0 = J 21 (ue + ); 0; 0, while the unprimed ones depend on ue; ; . These transformations simplify for the standard BMS group since the For the transformation law of e i0RI , we nd (u~ + ); 1 2 2Y 30R + Y 2 04Ri; 3Y 20R + 3Y 2 0 3R Y 3 04Ri; 4Y 10R + 6Y 2 0 2R 4Y 3 03R + Y 4 04Ri; dv @ 10R + 3 R0 02R 03R + (@ )2 04R ( ) : 02R + 3(@ )2 0 3R ) ; R00 = 000R = 3R 2 h 0 1R 1 f 0; g ; 2 04R; 3 2 h 0 2R 5 2 h 0 0R 3 3 2 2 0 e 2RI e 0RI Z 0 Z 0 ) : (6.107) The transformation laws of 02RI ; 01RI can be obtained from that of e 20RI ; e 01RI by using the rst two relations of (6.100) and equation (B.2), respectively (B.3) of appendix B. This gives 3 0 02RI = 1 0 2 Z u~ 0 dve _ R0 _ 0R ( { 34 { (6.104) (6.105) (6.106) (6.108) 0 01RI = 0 2R + 2 2 Z u~ 0 1 2 Z ve 0 dwe _ R0 _ 0R Z u~ 0 1 dv 3 0 R ( (6.109) Finally, the transformation law of the Bondi mass aspect as chosen in (6.101) is given by (1.3) after using (B.4). Note that the transformation law of the standard expression for the Bondi mass aspect can easily be obtained by using (B.1). 7 In this work, we have generalized nite BMS4 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{46]. The approach we have followed here is systematic and straightforward but explicit computations are rather tedious and can presumably be simpli ed 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 at solution space in the sense of Newman-Unti [27], containing not only the Kerr black e hole [47] but also Robinson-Trautman waves [48, 49]. In this context, the analog of the time coordinate 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 rst 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 e ect [51, 52] as described in [ 53 ] (see [ 54, 55 ] for recent discussions) is obvious. The question of what part of this e ect 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. Acknowledgments This work is supported in part by the Fund for Scienti c Research-FNRS (Belgium), by IISN-Belgium, and by \Communaute francaise de Belgique - Actions de Recherche ConD D D D D = ( = ( 2 ) 1 2 1 2 certees". C. Troessaert is Conicyt (Fondecyt postdoctoral grant 3140125) and Laurent Houart postdoctoral fellow. The Centro de Estudios Cient cos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt. The authors thank Pujian Mao for pointing out relevant references on the memory e ect. 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Glenn Barnich, Cédric Troessaert. Finite BMS transformations, Journal of High Energy Physics, 2016, 167, DOI: 10.1007/JHEP03(2016)167