# On the Nature of Bondi–Metzner–Sachs Transformations

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## Abstract

**:**

## 1. Introduction

- (i)
- The proof by F. Alessio and one of us [16] that the BMS group is the right semidirect product of the proper orthocronous Lorentz group ${\mathrm{SO}}^{+}(3,1)$ with supertranslations (cf. Appendix A).
- (ii)
- The division of BMS transformations into parabolic, elliptic, hyperbolic and loxodromic, since the first half of them consists of fractional linear maps which can be classified by studying their fixed points [17].
- (iii)
- The investigation of fractional linear maps in general relativity and quantum mechanics performed in Ref. [18].
- (iv)
- The proof that the BMS group is not real analytic, and the related suggestion that it is not locally exponential [19].
- (v)
- The recent discovery that groups of BMS type arise not only as macroscopic asymptotic symmetry groups in cosmology but describe also a fundamental microscopic symmetry of pseudo-Riemannian geometry [20].

## 2. Basic Framework

- (i)
**Parabolic.**Only one fixed point exists, for which ${(a+d)}^{2}=4$, while$$\mathsf{\Lambda}={A}_{P}=\left(\begin{array}{cc}\pm 1& \beta \\ 0& \pm 1\end{array}\right),$$$${f}_{\mathsf{\Lambda}}(\zeta )={f}_{P}(\zeta )=\zeta \pm \beta .$$- (ii)
**Elliptic.**Two fixed points exist, for which ${(a+d)}^{2}<4$, while$$\mathsf{\Lambda}={A}_{E}=\left(\begin{array}{cc}{e}^{i\frac{\chi}{2}}& 0\\ 0& {e}^{-i\frac{\chi}{2}}\end{array}\right),$$$${f}_{\mathsf{\Lambda}}(\zeta )={f}_{E}(\zeta )={e}^{i\chi}\zeta .$$- (iii)
**Hyperbolic.**Two fixed points exist, for which ${(a+d)}^{2}>4$, while$$\mathsf{\Lambda}={A}_{H}=\left(\begin{array}{cc}\sqrt{\left|\kappa \right|}& 0\\ 0& \frac{1}{\sqrt{\left|\kappa \right|}}\end{array}\right),$$$${f}_{\mathsf{\Lambda}}(\zeta )={f}_{H}(\zeta )=\left|\kappa \right|\zeta .$$- (iv)
**Loxodromic.**Two fixed points exist, for which ${(a+d)}^{2}\in \mathbb{C}-\mathbb{R}$, and$$(a+d)=\sqrt{k}+\frac{1}{\sqrt{k}},\phantom{\rule{0.277778em}{0ex}}k=\rho {e}^{i\sigma},\phantom{\rule{0.277778em}{0ex}}\rho \ne 1,$$

## 3. A New Theorem on Supertranslations

**Theorem**

**1.**

**Proof.**

## 4. Behavior of the Conformal Factor

## 5. Behavior of Killing Vector Fields under BMS Transformations

- (i)
**Parabolic.**In the case of a parabolic fractional linear map for $\zeta $$$\begin{array}{ccc}& & {K}_{P}(\zeta )=\frac{1+{\left|\zeta \right|}^{2}}{({1+|\pm \beta +\zeta |}^{2})},\hfill \end{array}$$$$\begin{array}{ccc}& & {\mathcal{N}}_{\gamma}={\mathcal{N}}_{P}=\pm 1,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\gamma ={\gamma}_{P}=\beta .\hfill \end{array}$$- (ii)
**Elliptic.**For an elliptic fractional linear map,$$\begin{array}{ccc}& & {K}_{E}(\zeta )=1,\hfill \end{array}$$$$\begin{array}{ccc}& & {\mathcal{N}}_{\gamma}={\mathcal{N}}_{E}=1,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\gamma ={\gamma}_{E}=0.\hfill \end{array}$$- (iii)
**Hyperbolic.**$$\begin{array}{ccc}& & {K}_{H}(\zeta )=\frac{\left|\kappa \right|(1+|\zeta {|}^{2})}{(1+{\left|\kappa \right|}^{2}{\left|\zeta \right|}^{2})},\hfill \end{array}$$$$\begin{array}{ccc}& & {\mathcal{N}}_{\gamma}={\mathcal{N}}_{H}=\left|k\right|,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\gamma ={\gamma}_{H}=0.\hfill \end{array}$$- (iv)
**Loxodromic.**$$\begin{array}{ccc}& & {K}_{L}(\zeta )=\frac{\rho (1+|\zeta {|}^{2})}{(1+{\rho}^{2}{\left|\zeta \right|}^{2})},\hfill \end{array}$$$$\begin{array}{ccc}& & {\mathcal{N}}_{\gamma}={\mathcal{N}}_{L}=\rho {e}^{i\sigma},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\gamma ={\gamma}_{L}=0.\hfill \end{array}$$

## 6. BMS Transformations in Homogeneous Coordinates

## 7. Concluding Remarks

- (i)
- Proof that to each normal elliptic transformation of the complex variable $\zeta $ used in the metric for cuts of null infinity, there corresponds a BMS supertranslation. Although this might be seen as a corollary of the work initiated in Ref. [17], it has prepared the ground for the items below.
- (ii)
- Study of the conformal factor in the BMS transformation of the u variable as a function of the squared modulus of $\zeta $. In the loxodromic and hyperbolic cases, such a conformal factor turns out to be either monotonically increasing or monotonically decreasing as a function of the real variable given by the absolute value of $\zeta $. In the parabolic case, the conformal factor is instead a real-valued function of complex variable, and one needs the plots of Figure 2.
- (iii)
- A classification of Killing vector fields of the Bondi metric has been obtained in Section 5.
- (iv)
- In Section 6, we have found that BMS transformations are the restriction to a pair of unit circles of a more general set of transformations. Within this broader framework, the geometry of such transformations is studied by means of its Segre manifold. This provides an unforeseen bridge between the language of algebraic geometry and the analysis of BMS transformations in general relativity.
- (v)
- Our remarks at the end of Section 5 might lead to a systematic application of projective geometry techniques for the definition of points at infinity in general relativity.
- (vi)
- Our results in Section 5 suggest four sets of Killing fields associated with the four branches of BMS transformations. As discussed in Section 3, the elliptic transformations (the case with ${K}_{\mathsf{\Lambda}}(\zeta )=1$) define the Abelian subgroup of supertranslations. The linearized action of supertranslations in the Schwarzschild case is already studed in [3], which results in a black hole with linearized supertranslation hair. It would be interesting to study the action of parabolic, hyperbolic and loxodromic transformations defined by the Killing fields (54) on a black hole metric.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Composition of BMS Transformations

## Appendix B. Origin and Properties of Fractional Linear Maps

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**Figure 1.**The conformal factor is monotonically increasing if $\mathsf{\Xi}\in ]0,1[$ and monotonically decreasing if $\mathsf{\Xi}>1$ in the hyperbolic and loxodromic cases.

**Figure 2.**${K}_{P}(\zeta )$ in the $(\zeta ,\overline{\zeta})$ plane. First row from left to right: ${K}_{P+}$ with $\beta $ as a real parameter, ${K}_{P+}$ with $\beta $ as a complex parameter. Second row: ${K}_{P+}$ with $\beta $ as a purely imaginary parameter, ${K}_{P-}$ with $\beta $ as a real parameter. Third row: ${K}_{P-}$ with $\beta $ as a complex parameter, ${K}_{P-}$ with $\beta $ as a purely imaginary parameter.

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Mirzaiyan, Z.; Esposito, G.
On the Nature of Bondi–Metzner–Sachs Transformations. *Symmetry* **2023**, *15*, 947.
https://doi.org/10.3390/sym15040947

**AMA Style**

Mirzaiyan Z, Esposito G.
On the Nature of Bondi–Metzner–Sachs Transformations. *Symmetry*. 2023; 15(4):947.
https://doi.org/10.3390/sym15040947

**Chicago/Turabian Style**

Mirzaiyan, Zahra, and Giampiero Esposito.
2023. "On the Nature of Bondi–Metzner–Sachs Transformations" *Symmetry* 15, no. 4: 947.
https://doi.org/10.3390/sym15040947