Superrotations as Charges: The Geometry of Conserved Quantities at the Boundary of Spacetime ✛ ✎
Context
Asymptotically flat spacetimes are described by the Bondi–Sachs metric, where symmetries at null infinity form the BMS group. The extension of this group by superrotations, proposed by Banks, Barnich, and Troessaert, faces a problem: the vector fields of superrotations diverge at the poles of the sphere, making the charges formally ill-defined. This work sits among recent attempts to regularize such divergences, from modified boundary conditions to phase-space renormalization. The authors return to Penrose's geometric method, previously applied to ordinary BMS symmetries, and extend it to superrotations.
Methods
The authors use Penrose's conformal compactification: the physical spacetime is embedded into an unphysical one with a boundary at null infinity. Charges are computed as integrals over cross-sections of this boundary using the Komar formula adapted by Geroch and Winicur. For superrotations, which are singular at the poles, the Flanagan–Nichols regularization is applied: the integration region is restricted to exclude neighborhoods of the poles, then the limit is taken. The authors verify that the integrand is finite despite inverse powers of the conformal factor, and that the gauge condition X=0 holds.
Results
It is shown that the Geroch–Winicour charge method applies to superrotations. The superrotation charge is expressed as an integral over a cross-section of null infinity, which becomes finite after regularization. A check with the example $Y^A$ = $z^2$ ∂_z yields a finite charge value. The charge is established to be covariant and to satisfy the continuity condition across cross-sections proposed by Chen, Wald, and Yau. The superrotation flux is also defined via a local formula that generalizes the result of Geroch and Winicour.
Implications
This work removes an obstacle to using superrotations as physical symmetries. Unlike neighboring approaches that rely on coordinate expansions or algebraic topology, this one uses Penrose's purely geometric method, which gives explicit covariance. This connects superrotations to the Wald–Zoupas program for defining conserved quantities in general relativity. The result is important for understanding the infrared structure of gravity and possible holography at null infinity.
Future development
The method can be extended to other extensions of the BMS group, for example to diffeomorphisms of the sphere or to superrotations in higher dimensions. It is interesting to investigate the connection with soft graviton theorems and the Virasoro algebra. Applications to black holes and to the analysis of gravitational wave memory are possible. The question of a local and covariant definition of the symplectic current at null infinity remains open.
Impact
The result impacts mathematical physics, gravitational theory, and holography. It provides a tool for computing charges in asymptotically flat spacetimes.
Next steps
Test the method on other examples of superrotations and on spacetimes with nonzero cosmological constant. Investigate the connection with the charge algebra and with quantization.
Key open problems
In general relativity, the problem of defining quasilocal conserved quantities remains unsolved. This work connects superrotations with continuity across cross-sections, which is a criterion for angular momentum. It is a step toward resolving the question of covariant charges at null infinity.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The authors apply Penrose's conformal completion method and the Geroch–Winicour linkage technique to compute superrotation charges and fluxes in asymptotically flat spacetime. The Flanagan–Nichols regularization renders the integrals finite, yielding a covariant definition of charges that is compatible with continuity across cross-sections.
Where its strength is
This work extends a well-established geometric method to superrotations, ensuring explicit covariance without coordinate expansions. It connects superrotations to the Wald–Zoupas program and paves the way for a rigorous definition of conserved quantities.
Where it could go next
- It might be interesting to compare the geometric superrotation charges with algebraic structures arising in tensor networks, which also model boundary symmetries and infrared modes.based on: Spacetime Bricks: How Tensor Networks Catch an Elusive Field Theory arXiv:2607.03534
- A related approach involves fuzzy probabilities in fuzzy spacetime; it might offer an alternative regularization of divergences at the poles of the celestial sphere if those poles are interpreted as fuzzy points.based on: Fuzzy Probabilities in a Blurry Spacetime arXiv:2605.23862
- Since superrotations are tied to the infrared structure of gravity, one could examine how analogous methods apply to hidden symmetries in the quantum realm, where infinite-dimensional algebras also emerge.based on: When Time and Space Swap Roles: The Hidden Symmetry of the Quantum World arXiv:2605.17507
Worth trying
- Investigate whether the superrotation charge is conserved when a cosmological constant is included, using the conformal completion technique for asymptotically de Sitter spacetimes.
- Explore the Poisson brackets of superrotation charges obtained via the geometric method and compare them with the BMS algebra extended by superrotations.
- Apply the Flanagan–Nichols regularization to other singular vector fields, such as supertranslations with singularities.
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The archive is sparse around this work: the closest review is only 0.49, against a usual 0.64. It looks like few have drilled nearby.
Superrotations include transformations that are not globally defined on the sphere: they diverge at a single point, like the function 1/z in the complex plane. This resembles the behavior of vortices in superfluid helium, where the singularity at the vortex core is compensated by circulation around it.
Key numbers
- spacetime dimension: 4
- number of smooth conformal Killing vectors on the sphere: 6
- number of generators of the Virasoro algebra: infinitely many
- order of Flanagan–Nichols regularization: exclusion of pole neighborhoods