Fully nonlinear transformations of the Weyl-Bondi-Metzner-Sachs asymptotic symmetry group

The asymptotic symmetry group of general relativity in asymptotically flat spacetimes can be extended from the Bondi-Metzner-Sachs (BMS) group to the generalized BMS (GMBS) group suggested by Campiglia and Laddha, which includes arbitrary diffeomorphisms of the celestial two-sphere. It can be further extended to the Weyl BMS (BMSW) group suggested by Freidel, Oliveri, Pranzetti and Speziale, which includes general conformal transformations. We compute the action of fully nonlinear BMSW transformations on the leading order Bondi-gauge metric functions: specifically, the induced metric, Bondi mass aspect, angular momentum aspect, and shear. These results generalize previous linearized results in the BMSW context by Freidel et al., and also nonlinear results in the BMS context by Chen, Wang, Wang and Yau. The transformation laws will be useful for exploring implications of the BMSW group.


Introduction
The systematic study of spacetimes that become asymptotically flat far from an isolated source was initiated in the 1960s.Bondi, van der Burg, and Metzner [1] as well as Sachs [2] used coordinates (now called Bondi coordinates) that are adapted to the outgoing null rays of an isolated gravitating and radiating system.Their analysis allowed them to compute the asymptotic Einstein equations in Bondi coordinates and the symmetry transformations that preserve the metric at infinity and the Bondi coordinate conditions.The group of such transformations, the Bondi-Metzner-Sachs (BMS) group, is larger than the Poincaré group of transformations that preserve Minkowski spacetime.Lorentz transformations are contained in the BMS group, and the group has a semi-direct product structure similar to that of the Poincaré group.However, the infinite-dimensional commutative group of supertranslations replaces the four-dimensional group of spacetime translations in the Poincaré group, though the spacetime translations remain a subgroup of the supertranslations.
More recently, the analysis of asymptotically flat spacetimes has been revisited, and larger groups of symmetry transformations were found which preserve different geometric quantities at future null infinity (this is described in more detail in Sec. 2) [3,4].These new groups include the extended BMS group [5,6], the generalized BMS group [7], and the Weyl-BMS (BMSW) group [8][9][10][11][12].While the action of BMS transformations on the Bondi metric functions has been computed to nonlinear order [13][14][15][16], the actions of the extended BMS, generalized BMS and BMSW transformations have been computed to linear order only [8,17].In this paper, we derive the nonlinear transformations of the leadingorder metric functions in vacuum for the generalized BMS and BMSW groups.These transformations reduce to the known nonlinear BMS and linearized BMSW results in the appropriate limits.
Fully nonlinear transformation laws can be useful for exploring the viability of alternative definitions of charges associated with symmetries.An example is the recent investigation of Ref. [18] of continuity of various charges as the cross-section of future null infinity is varied.They are also useful for understanding the space of vacua which is relevant to the quantum theory [8,17].
The remainder of this paper contains a review of the different asymptotic symmetry groups in Sec. 2. The main results are derived in Sec. 3 and compared with existing results in Sec. 4. Some applications of these results are given in Sec. 5.

Review of asymptotic symmetry groups in asymptotically flat spacetimes
In this section, we review the different asymptotic symmetry groups in asymptotically flat spacetime to establish our notation and conventions.We use retarded Bondi coordinates (u, r, θ 1 , θ 2 ) = (u, r, θ A ) near future null infinity, following Refs.[1,5,6,8,9,14,17,[19][20][21][22][23].We will use throughout the notations and conventions of Flanagan and Nichols [14] (henceforth FN).The key metric functions in Bondi gauge are the induced metric h AB (θ A ), the Bondi mass aspect m(u, θ A ), the angular momentum aspect N A (u, θ A ), and the shear C AB (u, θ A ).The metric expansion obtained from the asymptotic conditions, the gauge conditions and the vacuum Einstein equations is

1)
Here R is the two dimensional Ricci scalar of h AB (θ A ), and A, B are angular indices which run over the values 1, 2 and are raised and lowered with h AB and h AB , respectively.Also we have generalized the treatment of FN, following Compère, Fiorucci and Ruzziconi [17], to allow the induced metric h AB to differ from the canonical round metric, so that R is allowed to be an arbitrary function of θ A instead of being constrained to R = 2.This generalization requires replacing the leading term in the expansion (FN,2.3b)with R/2, adding the term D 2 R/8 to the right hand side of the evolution equation (FN,2.11a)for the Bondi mass aspect, and adding the term C AB D B R/4 to the evolution equation (FN,2.11b)for the angular momentum aspect [17].In this paper we specialize to spacetimes which are vacuum near I + , and so the subleading shear tensor D AB of (FN,2.3c)vanishes, by (FN,2.10).We will consider three different asymptotic groups obtained from three different phase space definitions (see Refs. [4,12] for reviews).In the coordinate system y i = (u, θ A ) on I + , the diffeomorphisms ψ : I + → I + have the following form for all three groups: where χ : S 2 → S 2 is a diffeomorphism of the two-sphere S 2 , and for a point P on I + we have defined y i = y i (P) and ȳi = y i (ψ(P)).The groups are: • The Weyl BMS (BMSW) group suggested by Freidel, Oliveri, Pranzetti and Speziale [8][9][10][11][12].For this group the two-sphere diffeomorphism χ and the functions β and α can be freely chosen.
• The generalized BMS (GBMS) group suggested by Campiglia and Laddha [7] and further studied in Refs.[17,[24][25][26][27][28][29].For this group the function α is determined as a function of χ as follows.Let ǫ AB be one of the two volume forms on the two-sphere that are determined up to sign by the metric h AB .Define the function ω χ by where χ * is the pullback operator.Then we have which will have the consequence that GBMS transformations preserve the volume form ǫ AB up to a sign (see Section 3.4 below).
• The BMS group [1,2,[30][31][32], the subgroup of GBMS for which the diffeomorphisms χ are restricted to be global conformal isometries of the two-sphere.As a consequence the metric h AB is preserved under BMS transformations.

Derivation of transformation laws
In this section, we derive the transformation properties of the metric functions under nonlinear BMSW transformations, for solutions which are vacuum near I + .

Supertranslations
Consider first supertranslations, and specifically the finite supertranslation ψ : Denoting the coordinates (u, θ A ) by y i and defining ȳi = y i • ψ = (ū, θA ) the mapping ψ on I + is given by ū Under this mapping the metric functions transform as Here dots denote derivatives with respect to u, D A is the covariant derivative associated with h AB , and R is the Ricci scalar of h AB .Also N AB = ĊAB is the Bondi news tensor, and (Dβ) 2 = D A βD A β.The transformation laws (3.2) apply to all three groups, BMSW, GBMS and BMS.
To derive the transformation laws (3.2), we extend the diffeomorphism (3.1) to a diffeomorphism η : M → M which coincides with ψ on I + , by assuming a general power series expansion in 1/r and using the notation x α = (u, r, θ A ) and xα = x α • η: Here the functions β (1) , β (2) , R (1) , R (2) , χ (1) , χ(2) and χ(3) are arbitrary.One subtlety is that although χ (1)A transforms as a vector under diffeomorphisms of the two-sphere S 2 , the higher order functions χ(2)A and χ(3)A in Eq. (3.3c) do not.We remedy this by parameterizing the diffeomorphism ϕ : S 2 → S 2 on the two sphere at fixed u and r in terms of three vector fields χ (1)A , χ (2)A and χ (3)A : Here for any vector field χ on S 2 the map ϕ χ (ε) is the knight diffeomorphism that moves any point ε units along the integral curve of χ that passes through that point [33,34].By comparing Eqs.(3.3c) and (3.4) we find that1 We next take the pullback ḡab = η * g ab of the metric (2.1) using the expansions (3.3) and imposing that it has the same form as the metric (2.1), but with different metric functions which we denote with overbars.We adopt the shorthand notation that O(αβ, n) means the O(r −n ) piece of the (αβ) component of this metric comparison.The derivation takes place in a series of steps where each step is simplified using results from the previous steps.From O(uu, 0), O(ur, 1), O(rA, 0), O(rr, 2) and the trace part of O(AB, −1), respectively, we find that Ṙ(1) = 0, (3.6a) β(1) = 0, (3.6b) We then obtain from O(AB, −2) and from the trace-free part of O(AB, −1) the transformation laws (3.2a) and (3.2b) for the metric h AB and shear tensor C AB .
We next take the pullback ḡab = η * g ab of the metric and follow the same steps as in Sec.3.1.From O(rA, 0), the trace part of O(AB, −1), and O(rr, 2), respectively, we find that We then obtain from O(AB, −2) and from the trace-free part of O(AB, −1) the transformation laws (3.9a) and (3.9b) for the metric h AB and shear tensor C AB .
Next we find from O(rA, 1), O(rr, 3), O(ru, 2) and O(rA, 2) respectively that Here the angular indices A, B are raised and lowered with h AB and not hAB given by Eq. (3.9a).Finally from O(uu, 1) and (uA, 1) we obtain the transformation laws (3.9c) and (3.9d) for the Bondi mass aspect m and angular momentum aspect N A .

Two-sphere diffeomorphisms of the Weyl BMS group
The third category of transformations in the BMSW group are diffeomorphisms of the two sphere: ū = u, θA = χ A (θ B ). (3.13)This extends to the exact four-dimensional diffeomorphism under which the metric functions transform by the pullback of the two-sphere diffeomorphism χ: (3.15d)

Two-sphere diffeomorphisms of the generalized BMS group
We now turn to the generalized BMS group instead of the Weyl BMS group, and focus again on two-sphere diffeomorphisms.The diffeomorphism ψ on I + takes the form Here the function α(θ A ) is determined as a function of χ by the requirement (2.5).
To compute the transformation of the metric functions under the mapping (3.16) we combine the results of Secs.3.2 and 3.3.We decompose ψ as ψ = ψ 2 • ψ 1 , where ψ 1 is the two-sphere diffeomorphism (3.13), and ψ 2 is the conformal transformation (3.8) with τ chosen to be τ = α • χ −1 . (3.17) The corresponding spacetime diffeomorphisms are related by η = η 2 • η 1 , and the pullback of the metric is then given by η It follows that we can compute transformed metric functions as follows.Start with the metric functions h AB , m, N A , C AB , and act with the pullback η 2 * which yields the transformed metric functions hAB , m, NA , CAB given by Eqs.(3.9) using the parameter (3.17).
Next, act with the pullback η 1 * using the prescription (3.15) which yields the final metric In particular from Eq. (3.9a) the final induced metric is given by ĥAB = χ * (e −2τ h AB ), and so the final volume form is where we have used Eqs.(3.17), (2.4) and (2.5).

Conformal isometries of the BMS group
We now turn to the boosts and rotations of the BMS group.These are a special case of the GBMS two-sphere diffeomorphisms (3.16)where χ is restricted to be a global conformal isometry, so that Here ω χ is defined by Eq. (2.4), and we are excluding improper Lorentz transformations.The transformation laws are therefore given by combining Eqs.(3.9), (3.17) and (3.19).It follows from the analysis that led to Eq. (3.20) that the two-metric is preserved, ĥAB = h AB .
Moreover for rotations we have ω χ = 1, and for boosts we have ω χ = (cosh γ−cos Θ sinh γ) 2 , where γ is the rapidity parameter and Θ is the angle between the boost direction and the direction determined by θ A .From Eqs. (2.4), (2.5) and (3.17) we obtain that It follows that e τ is purely l = 0 and l = 1, and so it is annihilated by the differential operator D A D B − h AB D 2 /2.This implies that the following terms vanish in Eqs.(3.9) for boosts and rotations: the second term in Eq. (3.9b), and the second, third and sixth terms in Eq. (3.9c).In addition the terms involving R in Eqs.(3.2) and (3.9) will vanish when we specialize to round two-metrics with R = 2, as is normally done for the BMS group.
• Our nonlinear computations in the BMS context given in Appendix B of Ref. [14] and Appendix B of Ref. [15].These method used in those computations was to combine the Bondi-coordinate charge expression (FN,3.5) of Ref. [14], which is known to be covariant, together with known nonlinear transformation of the symmetry generator vector fields on I + to indirectly deduce the transformations of the metric functions.
The results are limited to nonradiative regimes where N AB = 0 since the charge expression (FN,3.5)was derived only in that context.Demonstrating consistency with the results of this paper requires using (i) the condition N AB = 0; (ii) the simplifications discussed after Eq. (3.22) above; (iii) the definition (3.6) of Ref. [15] of the variable NA ; and (iv) the evolution equation (FN,2.11a)for the Bondi mass aspect which shows that ṁ = 0 in vacuum when N AB = 0.
• The nonlinear BMS transformation laws derived in Appendices C.5 and C.6 of the book [13] by Chrusciel, Jezierski, and Kijowski, using the fact that their angular momentum aspect is related to ours by a factor of −3.For supertranslations the results agree, except for the transformation law for N A , their Eq.(C.124).The difference seems to arise from a discrepancy between our Eq.(3.7b) for the function β (2) compared to their corresponding Eq. (C.114).For boosts the results do not agree; we have not been able to track down the source of the discrepancy in this case.
• The nonlinear BMS transformation laws recently derived by Chen, Wang, Wang and Yau in Ref. [16], which are consistent with our results when the simplifications discussed after Eq. (3.22) above are used.

Vacuum structure
One application of our nonlinear transformation results (3.2), (3.9), (3.15) and (3.19) is to obtain an explicit parameterization of "vacuum" states which in a local region of I + are diffeomorphic to the data for Minkowski spacetime.This allows us to reproduce and generalize slightly the GBMS results of Compère, Fiorucci and Ruzziconi given in Sec. 3 of Ref. [17].The result is as follows.First, following Refs.[8,9,17] we define the Liouville or Geroch tensor N vac AB [h CD ], a function of the metric h CD , as follows.We choose a conformal factor e 2τ to make e 2τ h AB a unit round two-metric, by solving the equation for τ .We then define the tensor which has the property General vacuum data can now be parameterized in terms of an arbitrary two-metric h AB (θ C ) and a function β(θ C ) as follows: ) N A = 0. (5.4d) The result (5.4), when restricted to metrics h AB whose volume form is fixed as appropriate for the GBMS group, agrees with Eqs.(3.10) and (3.26) of [17] 2 .The result (5.4) is also consistent with the results of Sec.4.7 of Ref. [8] on the vacuum structure.We obtain the form (5.4) of general vacuum data by applying a general BMSW transformation to Minkowski data.We parameterize the BMSW transformation by composing the two sphere diffeomorphism (3.13), the conformal transformation (3.8), and the supertranslation (3.1).We start with the Minkowski data hAB , CAB = 0, m = 0, NA = 0, (5.5)where hAB is a unit round two metric.Acting with the two-sphere diffeomorphism χ using Eq.Here DA and DA are the derivative operators associated with hAB and ĥAB and we used R = 2. From Eq. (5.7a) we can obtain generic two metrics ĥAB by choosing χ and τ appropriately, and it will be convenient following [17] to use ĥAB to parameterize the state rather than χ and τ .In the GBMS context of [17] the conformal transformation τ is constrained to be a function of χ, which constrains ĥAB to have the same volume form as hAB , as discussed in Sec.3.4 above.
The final step is to act with the supertranslation β using Eqs.(3.2).This yields .7d) where D 2 = D A D A and (Dβ) 2 = D A βD A β. Finally from O(uu, 1) and O(uA, 1) we obtain the transformation laws (3.2c) and (3.2d) for the Bondi mass aspect m and angular momentum aspect N A .