Homogeneous projective coordinates for the Bondi-Metzner-Sachs group

This paper studies the Bondi-Metzner-Sachs group in homogeneous projective coordinates, because it is then possible to write all transformations of such a group in a manifestly linear way. The 2-sphere metric, Bondi-Metzner-Sachs metric, asymptotic Killing vectors, generators of supertranslations, as well as boosts and rotations of Minkowski spacetime, are all re-expressed in homogeneous projective coordinates. Last, the integral curves of vector fields which generate supertranslations are evaluated in detail. This work prepares the ground for more advanced applications of the differential geometry of asymptotically flat spacetimes in projective coordinates.

The appropriate geometric framework can be summarized as follows.In spacetime models for which null infinity can be defined, the cuts of null infinity are spacelike twosurfaces orthogonal to the generators of null infinity [33].On using the familiar stereographic coordinate the first half of Bondi-Metzner-Sachs transformations read as where the matrix Λ = i.e., the group of fractional linear maps f Λ according to Eq. (1.2) with the associated matrix Λ.Since (1.5) The fractional linear maps (1.2) can be defined for all values of ζ upon requiring that Moreover, under fractional linear maps, lengths along the generators of null infinity scale according to where the conformal factor is given by [19,33] (1.8) By integration, Eq. (1.7) yields the second half of Bondi-Metzner-Sachs transformations: (1.9) As was pointed out in Ref. [19], the complex homogeneous coordinates associated to the Bondi-Metzner-Sachs transformation (1.2) have modulus ≤ 1, which is the equation of a unit circle, and are In other words, upon remarking that Eq. (1.2) is equivalent to the linear transformation law (1.12) The next step of the program initiated in Ref. [19] consists in realizing that, much in the same way as the affine transformations in the Euclidean plane can be re-expressed with the help of a 3 × 3 matrix in the form one can further re-express Eq. (1.12) with the help of a 3 × 3 matrix in the form with the understanding that Eq. (1.12) is the restriction to the unit circle Γ of the map (1.15), upon defining Within this extended framework, one can consider two complex projective planes [19].Let P be a point of the first plane with coordinates (w 0 , w 1 , w 2 ), and let P ′ be a point of the second plane, with coordinates (u 0 , u 1 , u 2 ).One can now consider the nine products between a complex coordinate of P and a complex coordinate of P ′ , i.e.
This equation provides the coordinate description of the Segre manifold [34,35], which is the projective image of the product of projective spaces.It is a natural tool for accommodating the transformations that reduce to the BMS transformations upon restriction to the unit circle Γ.It contains a complex double infinity of planes, two arrays of planes, and a complex fourfold infinity of quadrics [19,34], but its differential geometry is still largely unexplored, as far as we know.
Unlike Ref. [19], we have a more concrete task: since the Bondi-Metzner-Sachs transformation (1.2) becomes linear when expressed in terms of z 0 and z 1 , we are aiming to develop the Bondi-Metzner-Sachs formalism with the associated Killing vector fields by using the pair of variables (z 0 , z 2 Homogeneous coordinates on the 2-sphere It is useful, as an instrument to develop the BMS formalism in homogeneous coordinates, to re-write the 2-sphere metric in the desired coordinates.By using the definition (1.10), we get while for ϕ we obtain By virtue of the identity we obtain for ϕ the more convenient expression In order to re-express the 2-sphere metric, let us evaluate while Eventually, we obtain the metric for the 2-sphere in homogeneous coordinates At this stage, upon defining the real-valued function we can write the matrix of metric components in the form with non-vanishing determinant −16z 2 0 z 2 1 γ 2 and inverse matrix We can see from (2.1) that the terms are real-valued, whereas are complex.

Bondi-Sachs metric in homogeneous coordinates
We can now write the retarded Bondi-Sachs (hereafter BS) metric in homogeneous coordinates with the help of the previous formulae.For this purpose, let us first write the general BS metric in the form On passing from (θ, ϕ) to (z 0 , z 1 ) coordinates, we find the metric components of (3.1) expressed as follows (the material from our Eq.(3.2) to our Eq.(3.17) can be obtained from Eqs. (4.33), (4.35) and (4.37) in Ref. [36], which relies in turn upon the work in Ref. [37]): The Bondi gauge ∂ r det r −2 g AB = 0 implies that γ AB C AB = 0, where γ AB is given in Eq. (2.9).With our coordinates, this relation reads as We no longer have the simple result C zz = 0 for the mixed component as in the stereographic coordinates, because in homogeneous coordinates we obtain which implies that The angular components of the metric are where, of course, γ AB is given in Eq. (2.8).These formulae, jointly with the falloff conditions help to rewrite Upon assuming that β 1 /r ≪ 1, we get while for g uz 0 and g uz 1 we find and where use has been made of (3.8).Eventually, we get the matrix of Bondi metric components The gauge condition det g AB /r 2 = 0, instead of giving a solution for D AB such as in stereographic coordinates, gives us a condition for In order to determine the various coefficients in the falloff conditions, we require that the Bondi metric should satisfy the Einstein equations Upon restricting to the vacuum case T = 0, in the limit as r approaches ∞ in the Einstein tensor, first looking at G rr , and neglecting the terms of order O(r −4 ), we get Upon looking at G rz 0 and G rz 1 respectively, we get lengthy relations for U z 1 2 and U z 0 2 , compared to the stereographic coordinates case, which depend on other coefficients.However, we still manage to solve directly for U z 0 2 and U z 1 2 .On studying G rA = 0 we find and where we recall that C z 0 z 1 is given in Eq. (3.8).By virtue of Eqs.(3.12) and (3.13) we find eventually the metric in the form For the discussion of Bondi's news tensor, mass and angular momentum aspects we refer again to the work in Refs.[36,37].Now we are ready to evaluate the BMS generators in homogeneous coordinates in order to determine the supertranslations.

Asymptotic Killing fields
After finding the most general Bondi metric in homogeneous coordinates satisfying the asymptotically flat spacetime falloffs, our aim is to find the most general vector fields ξ satisfying the Bondi gauge condition and the asymptotically flat spacetime falloffs.As is well known, the Killing vectors solve by definition the equations Moreover, the preservation of the Bondi gauge condition yields (L ξ g) rr = 0, (L ξ g) rA = 0 and g AB (L ξ g) AB = 0. (4.1) From these relations one can calculate the four components of ξ µ .At this stage, we can compute the asymptotic Killing fields in homogeneous coordinates by using the familiar transformation law of vector fields.In other words, the work in Ref. [22] has defined the stereographic variable (we write ψ rather than z used in Ref. [22], in order to avoid confusion with our ζ in Eq. (1.1)) and has found, in Bondi coordinates u, r, θ, ϕ, the asymptotic Killing fields ξ + T where the components depend on a function f and on the Bondi coordinates.On denoting as usual by Y m l the spherical harmonics on the 2-sphere, one finds Now by virtue of the basic identities and upon exploiting the formulae (A7)-(A10) in the Appendix, we find ) ) where the values taken by the A ij functions can be read off from (4.9)- (4.11).At this stage, a patient evaluation proves that such vector fields have vanishing Lie brackets: The result is simple, but the actual proof requires several details, for which we refer the reader to Appendix B.

Flow of supertranslation vector fields
The analysis in this section does not have a direct impact on unsolved problems, but (as far as we can see) can help the general reader.More precisely, in order to appreciate that the familiar geometric constructions are feasible also in projective coordinates, we now consider the flow of supertranslation vector fields (4.9)-(4.11).For example, by virtue of (2.7), and defining p = (u, r, z 0 , z 1 ), the task of finding the flow of the supertranslation vector fields  (4.9), (4.10) and (4.11) consists of solving a system of nonlinear and coupled differential equations.For this purpose, we denote by σ, Σ, χ, respectively, the appropriate flow, and define Figure 7: Numerical evaluation of the integral curve for the supertranslation vector field (4.9).The initial conditions (5.14) are taken to be u = 0, r = 1, z 0 = e i π 4 cos π 12 , z 1 = e −i π 4 sin π 12 .In this particular case, the real parts meet at a single point.
where W = σ, Σ, χ, respectively, with components W 1 , W 2 , W 3 , W 4 .Hence we study the following coupled systems of nonlinear differential equations: ) ) ) ) ) ) ) with the initial conditions The resulting equations can only be solved numerically, to the best of our knowledge, and such solutions are displayed in Figures from 1 to 9. Since the desired solutions are complex-valued, we have displayed both real and imaginary parts, with three choices of initial conditions.

Concluding remarks and open problems
As far as we can see, the interest of our investigation lies in having shown that homogeneous projective coordinates lead to a fully computational scheme for all applications of the BMS group.This might pay off when more advanced properties will be studied.In particular, we have in mind the concept of superrotations [22,23] on the one hand, and the physical applications of the Segre manifold advocated in our Introduction and in Ref. [19].In other words, since our Eq.(1.15) contains Eq. (1.12), which in turn is just a re-expression of the BMS transformation (1.2), one might aim at embedding the study of BMS symmetries into the richer mathematical framework of complex analysis in several variables [38] and algebraic geometry.The exploitment of the complex analysis approach to algebraic geometry appears promising because the singular points of functions of several complex variables neous projective coordinates z 0 and z 1 have also been considered in Ref. [42], but in that case, upon writing .11)one finds that the x, y, z coordinates for the embedding of the 2-sphere in three-dimensional Euclidean space are given by The global spacetime translations of Minkowski spacetime can be first re-expressed in u, r, ξ, ξ coordinates, and read eventually, in terms of the asymptotic Killing fields (4.9)-(4.11), Explicitly, we find The boost (K i ) and rotation (J ij ) vector fields for Lorentz transformations in Minkowski spacetime can be written in u, r, ξ, ξ coordinates as is shown, for example, in Ref. [22].At that stage, by using again Eqs.(4.7), (4.8) and (A7)-(A10) we find


has unit determinant (ad − bc) = 1 and belongs therefore to the group SL(2, C).The resulting projective version of the special linear group can be defined as the space of pairs ) one can write that PSL(2, C) is the quotient space SL(2, C)/δ, where δ is the homeomorphism defined by δ(a, b, c, d) = (−a, −b, −c, −d).