BMS Symmetry: The Infinite-Dimensional Symmetry of Flat Spacetime

Historical Context: Beyond Poincaré

For decades, physicists assumed that the symmetry group of asymptotically flat spacetime at null infinity (ℐ⁺) was simply the Poincaré group—the ten-dimensional group of Lorentz transformations and translations that leaves Minkowski spacetime invariant. This intuition seemed reasonable: far from any gravitating matter, spacetime should look increasingly flat, and its symmetries should reduce to those of special relativity.

In 1962, Hermann Bondi, M. G. J. van der Burg, and A. W. K. Metzner discovered something remarkable. By carefully analyzing the boundary conditions at future null infinity in general relativity, they found that the actual symmetry group is vastly larger. Rainer Sachs independently arrived at similar conclusions. The resulting BMS group contains the Poincaré group as a finite-dimensional subgroup but extends it to include infinitely many additional transformations.

Mathematical Structure of BMS₄

Supertranslations

The defining feature of the BMS group is the replacement of ordinary spacetime translations with supertranslations. While Poincaré translations are parameterized by four constant parameters (one for each spacetime direction), supertranslations are parameterized by arbitrary smooth functions on the celestial sphere S².

Here u is the retarded time coordinate at null infinity, and f(z, z̄) is an arbitrary smooth function on the celestial sphere (with z denoting complex stereographic coordinates). This means there is one independent supertranslation for each point on the celestial sphere—infinitely many symmetry generators.

Superrotations

The BMS group can be further extended to include superrotations—angle-dependent generalizations of ordinary Lorentz transformations. These extend the Lorentz group SO(1,3) to the group of conformal transformations of the celestial sphere.

The full extended BMS group (sometimes called the generalized BMS group or BMSW group) includes both supertranslations and superrotations, forming an infinite-dimensional semi-direct product:

where Conf(S²) denotes the conformal group of the celestial sphere.

Physical Implications

Gravitational Memory

BMS supertranslations have a direct physical manifestation: gravitational memory effects. When a gravitational wave passes through a detector, it doesn't simply return the detector to its original state. Instead, there is a permanent displacement—a "memory" of the wave's passage.

This effect can be understood as a spontaneous breaking of BMS symmetry. The vacuum of quantum gravity is not invariant under supertranslations; instead, different supertranslation-related vacua correspond to different memory configurations. The Goldstone modes associated with this spontaneous symmetry breaking are precisely the gravitational memory observables.

Soft Graviton Theorems

One of the most beautiful connections in modern theoretical physics relates BMS symmetry to Weinberg's soft graviton theorem. In the 1960s, Steven Weinberg proved that scattering amplitudes with a very low-energy (soft) graviton factorize in a universal way.

Andrew Strominger and collaborators showed in 2013 that this soft theorem is mathematically equivalent to a Ward identity associated with BMS supertranslation symmetry. This means that the universal behavior of soft gravitons in quantum gravity is a direct consequence of the infinite-dimensional asymptotic symmetry:

Celestial Holography

In celestial holography, BMS symmetry plays a central organizing role. The 2D celestial conformal field theory (CFT) that describes 4D flat-space quantum gravity at null infinity has BMS as its fundamental symmetry group.

The celestial sphere S² serves as the base space for this holographic description, and scattering amplitudes in 4D momentum space are recast as correlation functions of operators on S². The BMS supertranslations act as global symmetries of this celestial CFT, while superrotations correspond to conformal transformations.

This provides a concrete realization of holography for asymptotically flat spacetimes, complementing the well-established AdS/CFT correspondence for anti-de Sitter spaces.

Current Research Frontiers

BMS Charges and Asymptotic Quantization

Canonical quantization of gravity at null infinity requires understanding the quantum algebra of BMS charges. Each supertranslation and superrotation generates a conserved charge at ℐ⁺, and these charges form an infinite-dimensional algebra that must be properly represented in the quantum theory.

Recent work has shown that the supertranslation charges have non-vanishing Dirac brackets, indicating that the phase space at null infinity is richer than previously appreciated. Understanding the quantum version of this algebra is essential for constructing a consistent quantum theory of gravity in flat space.

Generalizations to Other Dimensions

While BMS₄ refers specifically to four spacetime dimensions, analogous asymptotic symmetry groups exist in other dimensions. BMS₃ in three dimensions connects to two-dimensional Virasoro symmetry and has been extensively studied. Higher-dimensional generalizations (BMS_d for d > 4) exhibit increasingly complex structures and are active areas of research.

Connections to the S-Matrix

The existence of BMS symmetry has profound implications for the structure of the gravitational S-matrix. Unlike in theories with finite-dimensional symmetry groups, the infinite number of BMS charges provides infinitely many constraints on scattering amplitudes.

Understanding how to construct an S-matrix that respects all these Ward identities while maintaining unitarity and Lorentz invariance remains an active challenge. Some researchers have proposed that the S-matrix framework itself may need modification, perhaps replaced by a "celestial amplitude" formulation that makes BMS symmetry manifest from the start.

Conclusion

BMS symmetry represents a fundamental upgrade in our understanding of spacetime symmetries. The discovery that flat spacetime possesses infinitely more symmetries than the Poincaré group initially suggested has transformed our approach to quantum gravity in asymptotically flat spaces.

From gravitational memory effects that can in principle be measured by LIGO, to soft theorems that constrain all gravitational scattering, to the emerging framework of celestial holography, BMS symmetry provides a unifying thread connecting classical general relativity to quantum field theory to holographic duality.

As we continue to develop celestial holography as a viable approach to quantum gravity, BMS symmetry will undoubtedly remain at the center of the story—a reminder that even "empty" flat spacetime holds deep structural secrets waiting to be uncovered.