Homogeneous Projective Coordinates for the Bondi-Metzner-Sachs Group

1. Introduction

The Bondi-Metzner-Sachs [1,2,3] asymptotic symmetry group of asymptotically flat spacetime has received again much attention over the last decade by virtue of its relevance for black-hole physics [4,5,6], the group-theoretical structure of general relativity [7,8,9,10,11,12,13,14,15,16,17,18,19,20] and the infrared structure of fundamental interactions [21,22,23,24]. 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 two-surfaces orthogonal to the generators of null infinity [25]. On using the familiar stereographic coordinate

$$\zeta =e^{i\phi }cot\frac{\theta }{2},$$

(1)

the first half of Bondi-Metzner-Sachs transformations read as

$${\zeta }^{\prime }=f\left(\zeta \right)=\frac{(a\zeta +b)}{(c\zeta +d)}=f_{\Lambda }\left(\zeta \right),$$

(2)

where the matrix $\Lambda =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ has unit determinant $(ad-bc)=1$ and belongs therefore to the group $\mathrm{SL}(2,\mathbb{C})$. The resulting projective version of the special linear group can be defined as the space of pairs

$$\mathrm{PSL}(2,\mathbb{C})=\left\{(f,\Lambda )|f:\zeta \in \mathbb{C}\to f_{\Lambda }\left(\zeta \right),\Lambda \in \mathrm{SL}(2,\mathbb{C})\right\},$$

(3)

i.e., the group of fractional linear maps $f_{\Lambda }$ according to Eq. (2) with the associated matrix $\Lambda$. Since

$$f_{\Lambda }\left(\zeta \right)=\frac{(a\zeta +b)}{(c\zeta +d)}=\frac{(-a\zeta -b)}{(-c\zeta -d)}=f_{-\Lambda }\left(\zeta \right),$$

(4)

one can write that $\mathrm{PSL}(2,\mathbb{C})$ is the quotient space $\mathrm{SL}(2,\mathbb{C})/\delta$, where $\delta$ is the homeomorphism defined by

$$\delta (a,b,c,d)=(-a,-b,-c,-d).$$

(5)

The fractional linear maps (2) can be defined for all values of $\zeta$ upon requiring that

$$f_{\Lambda }(\infty )=\frac{a}{c},f_{\Lambda }\left(-\frac{d}{c}\right)=\infty .$$

(6)

Moreover, under fractional linear maps, lengths along the generators of null infinity scale according to

$$d{u}^{\prime }=K_{\Lambda }\left(\zeta \right)du,$$

(7)

where the conformal factor is given by [19,25]

$$K_{\Lambda }\left(\zeta \right)=\frac{1+{\left|\zeta \right|}^2}{{|a\zeta +b|}^2+{|c\zeta +d|}^2}.$$

(8)

By integration, Eq. (7) yields the second half of Bondi-Metzner-Sachs transformations:

$$u^{\prime }=K_{\Lambda }\left(\zeta \right)[u+\alpha (\zeta ,\overline{\zeta })].$$

(9)

As was pointed out in Ref. [19], the complex homogeneous coordinates associated to the Bondi-Metzner-Sachs transformation (2) have modulus $\le 1$, which is the equation of a unit circle, and are

$$z_0=e^{\frac{i\phi }{2}}cos\frac{\theta }{2},z_1=e^{-i\frac{\phi }{2}}sin\frac{\theta }{2}.$$

(10)

In other words, upon remarking that

$$\zeta =\frac{z_0}{z_1},$$

(11)

Eq. (2) is equivalent to the linear transformation law

$$\left(\begin{array}{c}{z}_0^{\prime }\\ z_1^{\prime }\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}{z}_0\\ z_1\end{array}\right).$$

(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

$$x^{\prime }=x+a,y^{\prime }=y+b,$$

(13)

can be re-expressed with the help of a $3\times 3$ matrix in the form

$$\left(\begin{array}{ccc}1& 0& a\\ 0& 1& b\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}x+a\\ y+b\\ 1\end{array}\right),$$

(14)

one can further re-express Eq. (12) with the help of a $3\times 3$ matrix in the form

$$\left(\begin{array}{c}{w}_0^{\prime }\\ w_1^{\prime }\\ w_2^{\prime }\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& a& b\\ 0& c& d\end{array}\right)\left(\begin{array}{c}{w}_0\\ w_1\\ w_2\end{array}\right),$$

(15)

with the understanding that Eq. (12) is the restriction to the unit circle $\Gamma$ of the map (15), upon defining

$${\left.w_0\right|}_{\Gamma }=1,{\left.w_1\right|}_{\Gamma }=z_0,{\left.w_2\right|}_{\Gamma }=z_1.$$

(16)

The work in Ref. [19] has outlined the resulting geometric picture, where $(w_0,w_1,w_2)$ are viewed as homogeneous coordinates in a complex projective plane. In our paper we have instead a less abstract and more concrete task: since the Bondi-Metzner-Sachs transformation (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 $(z_0,z_1)$ instead of $(\zeta ,\overline{\zeta })$. For this purpose, the homogeneous projective coordinates for the 2-sphere are studied in Sect. 2, while the Bondi-Sachs metric in homogeneous coordinates is considered in Sect. 3. Asymptotic Killing fields for supertranslations are evaluated in Sect. 4, while their flow is investigated in Sect. 5. Concluding remarks and open problems are presented in Sect. 6, while technical details are provided in the Appendices.

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 (10), we get

$$z_0{z}_1=sin(\theta /2)cos(\theta /2)={\displaystyle \frac{sin\left(\theta \right)}{2}}\Rightarrow \theta ={sin}^{-1}\left(2z_0{z}_1\right),$$

(17)

while for $\phi$ we obtain

$${\displaystyle \frac{z_0}{z_1}}=e^{i\phi }cot(\theta /2)\Rightarrow \phi =-ilog\left(tan(\theta /2){\displaystyle \frac{z_0}{z_1}}\right).$$

(18)

By virtue of the identity

$$tan(\theta /2)=={\displaystyle \frac{2sin(\theta /2)cos(\theta /2)}{2{cos}^2(\theta /2)}}={\displaystyle \frac{sin\left(\theta \right)}{1+\sqrt{1-{sin}^2\left(\theta \right)}}},$$

(19)

we obtain for $\phi$ the more convenient expression

$$\phi =-ilog\left(\frac{2z_0^2}{1+\sqrt{1-4z_0^2{z}_1^2}}\right).$$

In order to re-express the 2-sphere metric, let us evaluate

$$\begin{array}{ccc}\hfill d{\theta }^2& =& {\displaystyle \frac{d\theta }{d{z}_0}}{\displaystyle \frac{d\theta }{d{z}_0}}d{z}_0^2+{\displaystyle \frac{d\theta }{d{z}_1}}{\displaystyle \frac{d\theta }{d{z}_1}}d{z}_1^2+2{\displaystyle \frac{d\theta }{d{z}_0}}{\displaystyle \frac{d\theta }{d{z}_1}}d{z}_{0}d{z}_1\hfill \\ & =& \frac{4z_1^2}{1-4z_0^2{z}_1^2}d{z}_0^2+\frac{4z_0^2}{1-4z_0^2{z}_1^2}d{z}_1^2+\frac{8z_0{z}_1}{1-4z_0^2{z}_1^2}d{z}_{0}d{z}_1,\hfill \end{array}$$

(20)

while

$$\begin{array}{ccc}\hfill {sin}^2\left(\theta \right)d{\phi }^2& =& 4z_0^2{z}_1^{2}d{\phi }^2=4z_0^2{z}_1^2\left\{{\displaystyle \frac{d\phi }{d{z}_0}}{\displaystyle \frac{d\phi }{d{z}_0}}d{z}_0^2+{\displaystyle \frac{d\phi }{d{z}_1}}{\displaystyle \frac{d\phi }{d{z}_1}}d{z}_1^2+2{\displaystyle \frac{d\phi }{d{z}_0}}{\displaystyle \frac{d\phi }{d{z}_1}}d{z}_{0}d{z}_1\right\}\hfill \\ & =& {\displaystyle \frac{-16z_1^2{\left(1-2z_0^2{z}_1^2+\sqrt{1-4z_0^2{z}_1^2}\right)}^2}{{\left(1-4z_0^2{z}_1^2+\sqrt{1-4z_0^2{z}_1^2}\right)}^2}}d{z}_0^2-{\displaystyle \frac{64z_0^6{z}_1^4}{{\left(1-4z_0^2{z}_1^2+\sqrt{1-4z_0^2{z}_1^2}\right)}^2}}d{z}_1^2\hfill \\ & -& {\displaystyle \frac{32z_0^3{z}_1^3}{1-4z_0^2{z}_1^2}}d{z}_{0}d{z}_1.\hfill \end{array}$$

(21)

Eventually, we obtain the metric for the 2-sphere in homogeneous coordinates

$$\begin{array}{ccc}\hfill {\Omega }_2& =& d{\theta }^2+{sin}^2\left(\theta \right)d{\phi }^2=\sum _{\mu ,\nu =0}^1{g}_{\mu \nu }d{z}^{\mu }d{z}^{\nu }\hfill \\ & -& 4z_1^2\left(\frac{1-4z_0^2{z}_1^2+2\sqrt{1-4z_0^2{z}_1^2}}{1-4z_0^2{z}_1^2}\right)d{z}_0^2+8z_0{z}_{1}d{z}_{0}d{z}_1\hfill \\ & -& 4z_0^2\left(\frac{1-4z_0^2{z}_1^2-2\sqrt{1-4z_0^2{z}_1^2}}{1-4z_0^2{z}_1^2}\right)d{z}_1^2.\hfill \end{array}$$

(22)

At this stage, upon defining the real-valued function

$$\gamma (z_0,z_1)=\frac{2}{\sqrt{1-4z_0^2{z}_1^2}}=\frac{2}{cos\theta },$$

(23)

we can write the matrix of metric components in the form

$${\gamma }_{AB}=\left(\begin{array}{cc}-4z_1^2\left(1+\gamma \right)& 4z_0{z}_1\\ & \\ 4z_0{z}_1& -4z_0^2\left(1-\gamma \right)\end{array}\right),$$

(24)

with non-vanishing determinant $-16z_0^2{z}_1^2{\gamma }^2$ and inverse matrix

$${\gamma }^{AB}=\left(\begin{array}{cc}{\displaystyle \frac{1-\gamma }{4z_1^2{\gamma }^2}}& {\displaystyle \frac{1}{4z_0{z}_1{\gamma }^2}}\\ & \\ {\displaystyle \frac{1}{4z_0{z}_1{\gamma }^2}}& {\displaystyle \frac{1+\gamma }{4z_0^2{\gamma }^2}}\end{array}\right).$$

(25)

We can see from (17) that the terms

$$2z_0{z}_1=sin\left(\theta \right)\to 4z_0^2{z}_1^2={sin}^2\left(\theta \right)\to 1-4z_0^2{z}_1^2={cos}^2\left(\theta \right)\to 4z_0{z}_1=2sin\left(\theta \right),$$

are real-valued, whereas

$$z_0^2=e^{i\phi }{cos}^2(\theta /2),z_1^2=e^{-i\phi }{sin}^2(\theta /2)$$

are complex.

3. 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

$$d{s}^2=-Ud{u}^2-2e^{2\beta }dudr+h_{AB}\left(d{x}^A+\frac{1}{2}{U}^{A}du\right)\left(d{x}^B+\frac{1}{2}{U}^{B}du\right).$$

(26)

On passing from $(\theta ,\phi )$ to $(z_0,z_1)$ coordinates, we find the metric components of (3.1) expressed as follows:

$$g_{uu}=-U+\frac{1}{4}{h}_{z_0{z}_0}{\left(U^{z_0}\right)}^2+\frac{1}{4}{h}_{z_1{z}_1}{\left(U^{z_1}\right)}^2+\frac{1}{2}{h}_{z_0{z}_1}{U}^{z_0}{U}^{z_1},$$

(27)

$$g_{ur}=-e^{2\beta },$$

(28)

$$g_{u{z}_0}=\frac{1}{2}(h_{z_0{z}_0}{U}^{z_0}+h_{z_0{z}_1}{U}^{z_1}),$$

(29)

$$g_{u{z}_1}=\frac{1}{2}(h_{z_0{z}_1}{U}^{z_0}+h_{z_1{z}_1}{U}^{z_1}),$$

(30)

$$g_{z_0{z}_0}=h_{z_0{z}_0},g_{z_0{z}_1}=h_{z_0{z}_1},g_{z_1{z}_1}=h_{z_1{z}_1}.$$

(31)

The Bondi gauge ${\partial }_{r}det\left(r^{-2}{g}_{AB}\right)=0$ implies that [26] ${\gamma }^{AB}{C}_{AB}=0$, where ${\gamma }^{AB}$ is given in Eq. (2.9). With our coordinates, this relation reads as

$${\gamma }^{AB}{C}_{AB}=0\iff g^{z_0{z}_0}{C}_{z_0{z}_0}+g^{z_1{z}_1}{C}_{z_1{z}_1}+2g^{z_0{z}_1}{C}_{z_0{z}_1}=0.$$

We no longer have the simple result $C_{z\overline{z}}=0$ for the mixed component as in the stereographic coordinates, because in homogeneous coordinates we obtain

$${\displaystyle \frac{1-\gamma }{4z_1^2{\gamma }^2}}{C}_{z_0{z}_0}+{\displaystyle \frac{1+\gamma }{4z_0^2{\gamma }^2}}{C}_{z_1{z}_1}+{\displaystyle \frac{1}{2z_0{z}_1{\gamma }^2}}{C}_{z_0{z}_1}=0,$$

(32)

which implies that

$$C_{z_0{z}_1}=-{\displaystyle \frac{1}{2}}\left(1-\gamma \right){\displaystyle \frac{z_0}{z_1}}{C}_{z_0{z}_0}-{\displaystyle \frac{1}{2}}\left(1+\gamma \right){\displaystyle \frac{z_1}{z_0}}{C}_{z_1{z}_1}.$$

(33)

The angular components of the metric are

$$g_{z_0{z}_0}=r^2{\gamma }_{z_0{z}_0}+r{C}_{z_0{z}_0}+\mathcal{O}\left(r\right),g_{z_1{z}_1}=r^2{\gamma }_{z_1{z}_1}+r{C}_{z_1{z}_1}+\mathcal{O}\left(r\right),$$

$$\begin{array}{ccc}\hfill & & g_{z_0{z}_1}=r^2{\gamma }_{z_0{z}_1}+r{C}_{z_0{z}_1}+\mathcal{O}\left(r\right)\hfill \\ & & =r^2{\gamma }_{z_0{z}_1}-r\left({\displaystyle \frac{z_0}{z_1}}{\displaystyle \frac{\left(1-\gamma \right)}{2}}{C}_{z_0{z}_0}+{\displaystyle \frac{z_1}{z_0}}{\displaystyle \frac{\left(1+\gamma \right)}{2}}{C}_{z_1{z}_1}\right)+\mathcal{O}\left(r\right),\hfill \end{array}$$

where, of course, ${\gamma }_{AB}$ is given in Eq. (2.8). These formulae, jointly with the falloff conditions

$$\begin{array}{cc}\hfill & U(u,r,x^A)=1-{\displaystyle \frac{2m(u,r,x^A)}{r}}+{\displaystyle \frac{U_2(u,x^A)}{r^2}}+\mathcal{O}\left(r^{-3}\right)\hfill \\ \hfill & \beta (u,r,x^A)={\displaystyle \frac{{\beta }_1(u,x^A)}{r}}+{\displaystyle \frac{{\beta }_2(u,x^A)}{r^2}}+\mathcal{O}\left(r^{-3}\right)\hfill \\ \hfill & U^A(u,r,x^B)={\displaystyle \frac{U_2^A(u,x^B)}{r^2}}+{\displaystyle \frac{U_3^A(u,x^B)}{r^3}}+\mathcal{O}\left(r^{-4}\right)\hfill \\ \hfill & g_{AB}(u,r,x^A)=r^2{\gamma }_{AB}\left(x^A\right)+r{C}_{AB}(u,x^A)+D_{AB}(u,x^A)+\mathcal{O}\left(r^{-1}\right),\hfill \end{array}$$

(34)

help to rewrite

$$g_{uu}=-\left(1-{\displaystyle \frac{2m}{r}}\right)+\mathcal{O}\left(r^{-2}\right).$$

(35)

Upon assuming that ${\beta }_1/r\ll 1$, we get

$$g_{ur}=-\exp \left({\displaystyle \frac{2{\beta }_1}{r}}+\mathcal{O}\left(r^{-2}\right)\right)=-1-{\displaystyle \frac{2{\beta }_1}{r}}+\mathcal{O}\left(r^{-2}\right),$$

(36)

while for $g_{u{z}_0}$ and $g_{u{z}_1}$ we find

$$\begin{array}{cc}\hfill g_{u{z}_0}& ={\displaystyle \frac{1}{2}}\left(r^2{\gamma }_{z_0{z}_0}+r{C}_{z_0{z}_0}\right)\left({\displaystyle \frac{U_2^{z_0}}{r^2}}+{\displaystyle \frac{U_3^{z_0}}{r^3}}\right)+{\displaystyle \frac{1}{2}}\left\{r^2{\gamma }_{z_0{z}_1}-r\left[{\displaystyle \frac{z_0}{z_1}}{\displaystyle \frac{\left(1-\gamma \right)}{2}}{C}_{z_0{z}_0}\right.\right.\hfill \\ \hfill & \left.\left.+{\displaystyle \frac{z_1}{z_0}}{\displaystyle \frac{\left(1+\gamma \right)}{2}}{C}_{z_1{z}_1}\right]\right\}\left({\displaystyle \frac{U_2^{z_1}}{r^2}}+{\displaystyle \frac{U_3^{z_1}}{r^3}}\right)\hfill \\ \hfill & ={\displaystyle \frac{{\gamma }_{z_0{z}_0}}{2}}{U}_2^{z_0}+{\displaystyle \frac{{\gamma }_{z_0{z}_1}}{2}}{U}_2^{z_1}+{\displaystyle \frac{1}{r}}\left[{\displaystyle \frac{{\gamma }_{z_0{z}_0}}{2}}{U}_3^{z_0}+{\displaystyle \frac{C_{z_0{z}_0}}{2}}{U}_2^{z_0}+{\displaystyle \frac{{\gamma }_{z_0{z}_1}}{2}}{U}_3^{z_1}\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{z_0}{z_1}}{\displaystyle \frac{\left(1-\gamma \right)}{4}}{C}_{z_0{z}_0}{U}_2^{z_1}-{\displaystyle \frac{z_1}{z_0}}{\displaystyle \frac{\left(1+\gamma \right)}{4}}{C}_{z_1{z}_1}{U}_2^{z_1}\right]+\mathcal{O}\left(r^{-2}\right)\hfill \end{array}$$

(37)

and

$$\begin{array}{cc}\hfill g_{u{z}_1}& ={\displaystyle \frac{1}{2}}\left(r^2{\gamma }_{z_1{z}_1}+r{C}_{z_1{z}_1}\right)\left({\displaystyle \frac{U_2^{z_1}}{r^2}}+{\displaystyle \frac{U_3^{z_1}}{r^3}}\right)+{\displaystyle \frac{1}{2}}\left\{r^2{\gamma }_{z_0{z}_1}-r\left[{\displaystyle \frac{z_0}{z_1}}{\displaystyle \frac{\left(1-\gamma \right)}{2}}{C}_{z_0{z}_0}\right.\right.\hfill \\ \hfill & \left.\left.+{\displaystyle \frac{z_1}{z_0}}{\displaystyle \frac{\left(1+\gamma \right)}{2}}{C}_{z_1{z}_1}\right]\right\}\left({\displaystyle \frac{U_2^{z_0}}{r^2}}+{\displaystyle \frac{U_3^{z_0}}{r^3}}\right)\hfill \\ \hfill & ={\displaystyle \frac{{\gamma }_{z_1{z}_1}}{2}}{U}_2^{z_1}+{\displaystyle \frac{{\gamma }_{z_0{z}_1}}{2}}{U}_2^{z_0}+{\displaystyle \frac{1}{r}}\left[{\displaystyle \frac{{\gamma }_{z_1{z}_1}}{2}}{U}_3^{z_1}+{\displaystyle \frac{C_{z_1{z}_1}}{2}}{U}_2^{z_1}+{\displaystyle \frac{{\gamma }_{z_0{z}_1}}{2}}{U}_3^{z_0}\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{z_0}{z_1}}{\displaystyle \frac{\left(1-\gamma \right)}{4}}{C}_{z_0{z}_0}{U}_2^{z_0}-{\displaystyle \frac{z_1}{z_0}}{\displaystyle \frac{\left(1+\gamma \right)}{4}}{C}_{z_1{z}_1}{U}_2^{z_0}\right]+\mathcal{O}\left(r^{-2}\right),\hfill \end{array}$$

(38)

where use has been made of (3.8). Eventually, we get the matrix of Bondi metric components

$$g_{\mu \nu }=\left(\begin{array}{cccc}-\left(1-{\displaystyle \frac{2m}{r}}\right)& -1-{\displaystyle \frac{2{\beta }_1}{r}}& g_{u{z}_0}& g_{u{z}_1}\\ \\ -1-{\displaystyle \frac{2{\beta }_1}{r}}& 0& 0& 0\\ \\ g_{u{z}_0}& 0& r^2{\gamma }_{z_0{z}_0}+r{C}_{z_0{z}_0}& r^2{\gamma }_{z_0{z}_1}+r{C}_{z_0{z}_1}\\ \\ g_{u{z}_1}& 0& r^2{\gamma }_{z_0{z}_1}+r{C}_{z_0{z}_1}& r^2{\gamma }_{z_1{z}_1}+r{C}_{z_1{z}_1}\end{array}\right)+\mathcal{O}\left(r^{-2}\right).$$

(39)

The gauge condition $\det \left(g_{AB}/r^2\right)=0$, instead of giving a solution for $D_{AB}$ such as in stereographic coordinates, gives us a condition for $C_{AB}$

$$\det \left(g_{AB}\right)=\det \left(\begin{array}{cc}{r}^2{\gamma }_{z_0{z}_0}+r{C}_{z_0{z}_0}+D_{z_0{z}_0}& r^2{\gamma }_{z_0{z}_1}+r{C}_{z_0{z}_1}+D_{z_0{z}_1}\\ \\ r^2{\gamma }_{z_0{z}_1}+r{C}_{z_0{z}_1}+D_{z_0{z}_1}& r^2{\gamma }_{z_1{z}_1}+r{C}_{z_1{z}_1}+D_{z_1{z}_1}\end{array}\right)$$

$$\begin{array}{cc}\hfill & =r^4\left({\gamma }_{z_0{z}_0}{\gamma }_{z_1{z}_1}-{\gamma }_{z_0{z}_1}^2\right)+r^3\left({\gamma }_{z_0{z}_0}{C}_{z_1{z}_1}+{\gamma }_{z_1{z}_1}{C}_{z_0{z}_0}-2r^3{\gamma }_{z_0{z}_1}{C}_{z_0{z}_1}\right)\hfill \\ \hfill & +r^2\left({\gamma }_{z_0{z}_0}{D}_{z_1{z}_1}+C_{z_0{z}_0}{C}_{z_1{z}_1}+{\gamma }_{z_1{z}_1}{D}_{z_0{z}_0}-C_{z_0{z}_1}^2-2{\gamma }_{z_0{z}_1}{D}_{z_0{z}_1}\right)+\mathcal{O}\left(r\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \det \left({\displaystyle \frac{g_{AB}}{r^2}}\right)\Rightarrow & {\gamma }_{z_0{z}_0}{D}_{z_1{z}_1}+C_{z_0{z}_0}{C}_{z_1{z}_1}+{\gamma }_{z_1{z}_1}{D}_{z_0{z}_0}-C_{z_0{z}_1}^2-2{\gamma }_{z_0{z}_1}{D}_{z_0{z}_1}\stackrel{!}{=}0\hfill \\ \hfill & ⟹D_{z_0{z}_1}={\displaystyle \frac{{\gamma }_{z_0{z}_0}{D}_{z_1{z}_1}+C_{z_0{z}_0}{C}_{z_1{z}_1}+{\gamma }_{z_1{z}_1}{D}_{z_0{z}_0}-C_{z_0{z}_1}^2}{2{\gamma }_{z_0{z}_1}}}\hfill \\ \hfill & C_{z_0{z}_1}^2=C_{z_0{z}_0}{C}_{z_1{z}_1}.\hfill \end{array}$$

In order to determine the various coefficients in the falloff conditions, we require that the Bondi metric should satisfy the Einstein equations

$$G_{\mu \nu }\equiv R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }=8\pi G{T}_{\mu \nu }.$$

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 $\mathcal{O}\left(r^{-4}\right)$, we get

$$G_{rr}=-{\displaystyle \frac{4{\beta }_1}{r^3}}+\mathcal{O}\left(r^{-4}\right)\stackrel{!}{=}0\Rightarrow {\beta }_1\equiv 0.$$

Upon looking at $G_{r{z}_0}$ and $G_{r{z}_1}$ respectively, we get lengthy relations for $U_2^{z_1}$ and $U_2^{z_0}$, compared to the stereographic coordinates case, which depend on other coefficients. However, we still manage to solve directly for $U_2^{z_0}$ and $U_2^{z_1}$. On studying $G_{rA}=0$ we find

$$U_2^{z_0}={\displaystyle \frac{2z_0{z}_1\left(C_{z_1{z}_1}{U}_2^{z_1}+{\gamma }_{z_0{z}_1}{U}_3^{z_0}+{\gamma }_{z_1{z}_1}{U}_3^{z_1}\right)}{z_1^2\left(1+\gamma \right)C_{z_1{z}_1}+z_0^2\left(1-\gamma \right)C_{z_0{z}_0}}}=-{\displaystyle \frac{C_{z_1{z}_1}{U}_2^{z_1}+2{\gamma }_{z_0{z}_1}{U}_3^{z_0}}{C_{z_0{z}_1}}},$$

(40)

and

$$U_2^{z_1}={\displaystyle \frac{2z_0{z}_1\left(C_{z_0{z}_0}{U}_2^{z_0}+{\gamma }_{z_0{z}_0}{U}_3^{z_0}+{\gamma }_{z_0{z}_1}{U}_3^{z_1}\right)}{z_1^2\left(1+\gamma \right)C_{z_1{z}_1}+z_0^2\left(1-\gamma \right)C_{z_0{z}_0}}}=-{\displaystyle \frac{C_{z_0{z}_0}{U}_2^{z_0}+2{\gamma }_{z_0{z}_1}{U}_3^{z_1}}{C_{z_0{z}_1}}},$$

(41)

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

$$\begin{array}{ccc}\hfill & & d{s}^2=-d{u}^2-2dudr+2\left(r^2{\gamma }_{z_0{z}_1}+r{C}_{z_0{z}_1}\right)d{z}_{0}d{z}_1+{\displaystyle \frac{2m}{r}}d{u}^2+\left(r^2{\gamma }_{z_0{z}_0}+r{C}_{z_0{z}_0}\right)d{z}_0^2\hfill \\ & & +\left(r^2{\gamma }_{z_1{z}_1}+r{C}_{z_1{z}_1}\right)d{z}_1^2\hfill \\ & & +[\frac{{\gamma }_{z_0{z}_0}}{2}{U}_2^{z_0}+\frac{{\gamma }_{z_0{z}_1}}{2}{U}_2^{z_1}+\frac{1}{r}(\frac{C_{z_0{z}_0}}{2}{U}_2^{z_0}+\frac{C_{z_0{z}_1}}{2}{U}_2^{z_1}+\frac{{\gamma }_{z_0{z}_0}}{2}{U}_3^{z_0}+\frac{{\gamma }_{z_0{z}_1}}{2}{U}_3^{z_1}\left)\right]dud{z}_0\hfill \\ & & +[\frac{{\gamma }_{z_0{z}_1}}{2}{U}_2^{z_0}+\frac{{\gamma }_{z_1{z}_1}}{2}{U}_2^{z_1}+\frac{1}{r}(\frac{C_{z_0{z}_1}}{2}{U}_2^{z_0}+\frac{C_{z_1{z}_1}}{2}{U}_2^{z_1}+\frac{{\gamma }_{z_0{z}_1}}{2}{U}_3^{z_0}+\frac{{\gamma }_{z_1{z}_1}}{2}{U}_3^{z_1}\left)\right]dud{z}_1\hfill \\ & & +\mathcal{O}\left(r^{-2}\right).\hfill \end{array}$$

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Now we are ready to evaluate the BMS generators in homogeneous coordinates in order to determine the supertranslations.

4. 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 $\xi$ satisfying the Bondi gauge condition and the asymptotically flat spacetime falloffs. As is well known, the Killing vectors solve by definition the equations

$${\left({\mathcal{L}}_{\xi }g\right)}_{\mu \nu }={\xi }^{\rho }{\partial }_{\rho }{g}_{\mu \nu }+g_{\mu \rho }{\partial }_{\nu }{\xi }^{\rho }+g_{\nu \rho }{\partial }_{\mu }{\xi }^{\rho }=0.$$

Moreover, the preservation of the Bondi gauge condition yields [26]

$${\left({\mathcal{L}}_{\xi }g\right)}_{rr}=0,{\left({\mathcal{L}}_{\xi }g\right)}_{rA}=0\mathrm{and}{g}^{AB}{\left({\mathcal{L}}_{\xi }g\right)}_{AB}=0.$$

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From these relations one can calculate the four components of ${\xi }^{\mu }$. At this stage, instead of repeating the detailed calculations already available, for example, in Ref. [26], 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. [26] has defined the stereographic variable (we write $\psi$ rather than z used in Ref. [26], in order to avoid confusion with our $\zeta$ in Eq. (1.1))

$$\psi =e^{i\phi }tan\frac{\theta }{2}=\frac{1}{\overline{\zeta }},$$

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and has found, in Bondi coordinates $u,r,\theta ,\phi$, the asymptotic Killing fields ${\xi }_T^{+}$ where the components depend on a function f and on the Bondi coordinates. On denoting as usual by $Y_l^m$ the spherical harmonics on the 2-sphere, one finds [26]

$${\xi }_T^{+}{|}_{f=Y_0^0}=\frac{\partial }{\partial u},$$

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$${\xi }_T^{+}{|}_{f=Y_1^0}=\frac{(1-\psi \overline{\psi })}{(1+\psi \overline{\psi })}\left(\frac{\partial }{\partial u}-\frac{\partial }{\partial r}\right)+\frac{\psi }{r}\frac{\partial }{\partial \psi }+\frac{\overline{\psi }}{r}\frac{\partial }{\partial \overline{\psi }},$$

(46)

$${\xi }_T^{+}{|}_{f=Y_1^1}=\frac{\psi }{(1+\psi \overline{\psi })}\left(\frac{\partial }{\partial u}-\frac{\partial }{\partial r}\right)+\frac{{\psi }^2}{2r}\frac{\partial }{\partial \psi }-\frac{1}{2r}\frac{\partial }{\partial \overline{\psi }},$$

(47)

$${\xi }_T^{+}{|}_{f=Y_1^{-1}}=\frac{\overline{\psi }}{(1+\psi \overline{\psi })}\left(\frac{\partial }{\partial u}-\frac{\partial }{\partial r}\right)-\frac{1}{2r}\frac{\partial }{\partial \psi }+\frac{{\overline{\psi }}^2}{2r}\frac{\partial }{\partial \overline{\psi }}.$$

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Now by virtue of the basic identities

$$\frac{\partial }{\partial \psi }=\frac{\partial z_0}{\partial \psi }\frac{\partial }{\partial z_0}+\frac{\partial z_1}{\partial \psi }\frac{\partial }{\partial z_1},$$

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$$\frac{\partial }{\partial \overline{\psi }}=\frac{\partial z_0}{\partial \overline{\psi }}\frac{\partial }{\partial z_0}+\frac{\partial z_1}{\partial \overline{\psi }}\frac{\partial }{\partial z_1},$$

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and upon exploiting the formulae (A7)-(A10) in the Appendix, we find

$${\xi }_T^{+}{|}_{f=Y_1^0}=\frac{2}{\gamma }\left(\frac{\partial }{\partial u}-\frac{\partial }{\partial r}\right)+\frac{z_0}{2r}\frac{(2-\gamma )}{\gamma }\frac{\partial }{\partial z_0}+\frac{z_1}{2r}\frac{(2+\gamma )}{\gamma }\frac{\partial }{\partial z_1},$$

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$$\begin{array}{ccc}\hfill {\xi }_T^{+}{|}_{f=Y_1^1}& =& \frac{z_0}{2z_1}\frac{(\gamma -2)}{\gamma }\left(\frac{\partial }{\partial u}-\frac{\partial }{\partial r}\right)+\frac{1}{r}\frac{{\left(z_0\right)}^2}{z_1}\left(\frac{1}{4}-\frac{1}{\gamma (\gamma +2)}\right)\frac{\partial }{\partial z_0}\hfill \\ & +& \frac{z_0}{2r}\left(\frac{1}{(\gamma +2)}-\frac{(\gamma +2)}{2\gamma }\right)\frac{\partial }{\partial z_1},\hfill \end{array}$$

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$$\begin{array}{ccc}\hfill {\xi }_T^{+}{|}_{f=Y_1^{-1}}& =& \frac{z_1}{2z_0}\frac{(\gamma +2)}{\gamma }\left(\frac{\partial }{\partial u}-\frac{\partial }{\partial r}\right)-\frac{z_1}{2r}\left(\frac{1}{(\gamma -2)}+\frac{(\gamma -2)}{2\gamma }\right)\frac{\partial }{\partial z_0}\hfill \\ & +& \frac{1}{r}\frac{{\left(z_1\right)}^2}{z_0}\left(\frac{1}{4}-\frac{1}{\gamma (\gamma -2)}\right)\frac{\partial }{\partial z_1}.\hfill \end{array}$$

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Now we denote by ${\xi }_0,{\xi }_1,{\xi }_2,{\xi }_3$ the vector fields (4.3), (4.9), (4.10) and (4.11), respectively. Nontrivial Lie brackets among them involve ${\xi }_1,{\xi }_2,{\xi }_3$ only. With our notation, we can re-write Eqs. (4.9)-(4.11) in the form

$${\xi }_1=A_{11}\left(\frac{\partial }{\partial u}-\frac{\partial }{\partial r}\right)+A_{12}\frac{\partial }{\partial z_0}+A_{13}\frac{\partial }{\partial z_1},$$

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$${\xi }_2=A_{21}\left(\frac{\partial }{\partial u}-\frac{\partial }{\partial r}\right)+A_{22}\frac{\partial }{\partial z_0}+A_{23}\frac{\partial }{\partial z_1},$$

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$${\xi }_3=A_{31}\left(\frac{\partial }{\partial u}-\frac{\partial }{\partial r}\right)+A_{32}\frac{\partial }{\partial z_0}+A_{33}\frac{\partial }{\partial z_1},$$

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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:

$$[{\xi }_1,{\xi }_2]=[{\xi }_2,{\xi }_3]=[{\xi }_3,{\xi }_1]=0.$$

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The result is simple, but the actual proof requires several details, for which we refer the reader to Appendix B.

5. Flow of Supertranslation Vector Fields

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 $\sigma ,\Sigma ,\chi$, respectively, the appropriate flow, and define

$$\delta (W;\tau ,p)=\sqrt{1-4(W^3(\tau ,p)W^4(\tau ,p){)}^2},$$

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where $W=\sigma ,\Sigma ,\chi$, respectively, with components $W^1,W^2,W^3,W^4$. Hence we study the following coupled systems of nonlinear differential equations:

$$\frac{d{\sigma }^1}{d\tau }=\delta (\sigma ;\tau ,p),$$

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$$\frac{d{\sigma }^2}{d\tau }=-\delta (\sigma ;\tau ,p),$$

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$$\frac{d{\sigma }^3}{d\tau }=\frac{{\sigma }^3(\tau ,p)}{2{\sigma }^2(\tau ,p)}(\delta (\sigma ;\tau ,p)-1),$$

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$$\frac{d{\sigma }^4}{d\tau }=\frac{{\sigma }^4(\tau ,p)}{2{\sigma }^2(\tau ,p)}(\delta (\sigma ;\tau ,p)+1),$$

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$$\frac{d{\Sigma }^1}{d\tau }=\frac{{\Sigma }^3(\tau ,p)}{2{\Sigma }^4(\tau ,p)}(1-\delta (\Sigma ;\tau ,p)),$$

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$$\frac{d{\Sigma }^2}{d\tau }=-\frac{{\Sigma }^3(\tau ,p)}{2{\Sigma }^4(\tau ,p)}(1-\delta (\Sigma ;\tau ,p)),$$

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$$\frac{d{\Sigma }^3}{d\tau }=\frac{{\left({\Sigma }^3(\tau ,p)\right)}^2}{4{\Sigma }^2(\tau ,p){\Sigma }^4(\tau ,p)}\left[1-\delta (\Sigma ;\tau ,p)+\frac{\delta (\Sigma ;\tau ,p)}{(1+\delta (\Sigma ;\tau ,p\left)\right)}\right],$$

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$$\frac{d{\Sigma }^4}{d\tau }=\frac{{\Sigma }^3(\tau ,p)}{4{\Sigma }^2(\tau ,p)}\left[\frac{\delta (\Sigma ;\tau ,p)}{(1+\delta (\Sigma ;\tau ,p\left)\right)}-1-\delta (\Sigma ;\tau ,p)\right],$$

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$$\frac{d{\chi }^1}{d\tau }=\frac{{\chi }^4(\tau ,p)}{2{\chi }^3(\tau ,p)}(1+\delta (\chi ;\tau ,p)),$$

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$$\frac{d{\chi }^2}{d\tau }=-\frac{{\chi }^4(\tau ,p)}{2{\chi }^3(\tau ,p)}(1+\delta (\chi ;\tau ,p)),$$

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$$\frac{d{\chi }^3}{d\tau }=-\frac{{\chi }^4(\tau ,p)}{4{\chi }^2(\tau ,p)}\left[\frac{\delta (\chi ;\tau ,p)}{(1-\delta (\chi ;\tau ,p)}+1-\delta (\chi ;\tau ,p)\right],$$

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$$\frac{d{\chi }^4}{d\tau }=\frac{{\left({\chi }^4(\tau ,p)\right)}^2}{4{\chi }^2(\tau ,p){\chi }^3(\tau ,p)}\left[1+\delta (\chi ;\tau ,p)-\frac{\delta (\chi ;\tau ,p)}{(1-\delta (\chi ;\tau ,p\left)\right)}\right],$$

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with the initial conditions

$$W^1(0,p)=u,W^2(0,p)=r,W^3(0,p)=z_0,W^4(0,p)=z_1.$$

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The resulting equations can only be solved numerically, to the best of our knowledge, and such solutions are displayed in Figure 1,Figure 2,Figure 3,Figure 4,Figure 5,Figure 6,Figure 7,Figure 8,Figure 9. Since the desired solutions are complex-valued, we have displayed both real and imaginary parts, with three choices of initial conditions.

6. 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 [21,22,26] on the one hand, and the physical applications of the Segre manifold advocated 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 [27] and algebraic geometry. Our paper has tried to prepare the ground for such a synthesis.