On the structure and applications of the Bondi-Metzner-Sachs group

TL;DR

This work reviews the Bondi-Metzner-Sachs (BMS) group as the asymptotic symmetry group of asymptotically flat spacetimes, combining Bondi-Sachs coordinates with Penrose conformal infinity to elucidate the structure and representations of $\mathscr{B}=\mathscr{L}\rtimes\mathscr{S}$. It details the geometric origin of BMS, its Lie algebra, and the distinction between good cuts and the general lack of a canonical Poincaré subgroup in generic spacetimes, while connecting these symmetries to gravitational scattering and energy loss via the Bondi mass and news function. The discussion extends to CK spacetimes, the linking of $\mathscr{I}^+$ and $\mathscr{I}^-$ near spacelike infinity, and the contemporary implications for black hole physics and quantum gravity, including soft hair and infrared structures. Overall, the paper integrates conformal methods, asymptotic analysis, and scattering theory to illuminate how asymptotic symmetries govern gravitational radiation and its quantum implications.

Abstract

This work is a pedagogical review dedicated to a modern description of the Bondi-Metzner-Sachs group. The curved space-times that will be taken into account are the ones that suitably approach, at infinity, Minkowski space-time. In particular we will focus on asymptotically flat space-times. In this work the concept of asymptotic symmetry group of those space-times will be studied. In the first two sections we derive the asymptotic group following the classical approach which was basically developed by Bondi, van den Burg, Metzner and Sachs. This is essentially the group of transformations between coordinate systems of a certain type in asymptotically flat space-times. In the third section the conformal method and the notion of asymptotic simplicity are introduced, following mainly the works of Penrose. This section prepares us for another derivation of the Bondi-Metzner-Sachs group which will involve the conformal structure, and is thus more geometrical and fundamental. In the subsequent sections we discuss the properties of the Bondi-Metzner-Sachs group, e.g. its algebra and the possibility to obtain as its subgroup the Poincaré group, as we may expect. The paper ends with a review of the Bondi-Metzner-Sachs invariance properties of classical gravitational scattering discovered by Strominger, that are finding application to black hole physics and quantum gravity in the literature.

Figures (9)

• Figure 1: The Bondi-Sachs coordinate system. The coordinates $u$, $r$, and $\phi$ and the vector $k^a$ are shown in the hypersurface $\theta=\mathrm{const}$.
• Figure 2: The cylinder $\mathscr{E}=S^3\times\mathbb{R}$, of which $\mathscr{M}$ is just a finite portion, delimited by $\mathscr{I^+}$, $\mathscr{I^-}$, $i^+$, $i^-$ and $i^0$. We note that the $(\theta,\phi)$ coordinates are suppressed, so that each point represents a 2-sphere of radius $\sin r'$.
• Figure 3: A Penrose diagram for $\mathscr{M}$, using $(t',r')$ coordinates.
• Figure 4: This is another useful way of depicting $\mathscr{M}$ as the interior of two cones joined base to base. This picture however is not conformally accurate: in fact $i^0$ appears as an equatorial region whereas it should be a point.
• Figure 5: Null infinity for Schwarzschild space-time. Note that $w=0$ corresponds both to $\mathscr{I^+}$ and $\mathscr{I^-}$. The points $i^{\pm}$ and $i^0$ are singular and have been deleted.

Theorems & Definitions (33)

• Definition 2.1
• Definition 3.1
• Proposition 3.1
• Definition 3.2
• Remark 3.1
• Definition 4.1
• Definition 4.2
• Remark 4.1
• Remark 4.2
• Definition 4.3