Lectures on the Bondi--Metzner--Sachs group and related topics in infrared physics

TL;DR

This work develops the Bondi–Metzner–Sachs framework for asymptotic symmetries of null infinity in four-dimensional, asymptotically flat spacetimes. It builds the BMS group from Carrollian geometry and conformal Killing structures, provides dual derivations (conformal-compactification and Bondi–Sachs), and connects these symmetries to the Weinberg soft graviton theorem, gravitational memory, and Hadamard-state construction. The notes also discuss generalizations like superrotations, the infrared triangle, and electrodynamic analogs, including large gauge transformations and memory. The synthesis clarifies how asymptotic symmetries govern infrared physics, scattering, and quantum structure of gravity, with implications for black-hole hair and celestial holography.

Abstract

These are the extended lecture notes for a minicourse presented at the I São Paulo School on Gravitational Physics discussing the Bondi--Metzner--Sachs (BMS) group, the group of symmetries at null infinity on asymptotically flat spacetimes. The BMS group has found many applications in classical gravity, quantum field theory in flat and curved spacetimes, and quantum gravity. These notes build the BMS group from its most basic prerequisites (such as group theory, symmetries in differential geometry, and asymptotic flatness) up to modern developments. These include its connections to the Weinberg soft graviton theorem, the memory effect, its use to construct Hadamard states in quantum field theory in curved spacetimes, and other ideas. Advanced sections briefly discuss the main concepts behind the infrared triangle in electrodynamics, superrotations, and the Dappiaggi--Moretti--Pinamonti group in expanding universes with cosmological horizons. New contributions by the author concerning asymptotic (conformal) Killing horizons are discussed at the end.

Figures (20)

• Figure 1: Diagram for construction of stereographic coordinates. One picks a point on the sphere and connects it to the north pole. One follows the line determined in this manner until it intercepts the $xy$-plane, at which point one records the coordinates $x$ and $y$ of the intersection. The coordinates attributed to the point on the sphere are then $\zeta = x + iy$ and its conjugate $\bar{\zeta}$. This corresponds to $\zeta = e^{i\phi}\cot(\frac{\theta}{2})$.
• Figure 2: Illustration, with one dimension suppressed, of the physical meaning of the null coordinates $u$ and $v$. Surfaces of constant $u$ are outgoing spherical "waves", while surfaces of constant $v$ are their incoming analogues. This figure is based on Fig. 12.i of Ref. hawking1973LargeScaleStructure.
• Figure 3: Carter--Penrose diagram of Minkowski spacetime.
• Figure 4: Minkowski spacetime embedded in the Einstein static universe (two spatial dimensions are suppressed). Since the Einstein static universe has the topology $\mathbb{R} \times \mathbb{S}^3$, it is often depicted as a cylinder. The Carter--Penrose diagram for Minkowski spacetime is wrapped around the Einstein cylinder.
• Figure 5: Different curves represented in the Carter--Penrose diagram for Minkowski spacetime. Vertical curves (far-left) are curves with constant $r$. Horizontal curves (center-left) are curves with constant $t$. Diagonal lines have constant $u$ (outgoing, center-right) or $v$ (incoming, far-right).

Theorems & Definitions (25)

• Definition 1: Group
• Definition 2: Subgroup
• Definition 3: Lie Group
• Definition 4: Homomorphism
• Definition 5: Isomorphism
• Definition 6: Normal Subgroup
• Definition 7: Semidirect Product of Groups
• Definition 8: Direct Product of Groups
• Definition 9: Immersion
• Definition 10: Embedding