Cosmic footballs from superrotations

TL;DR

The paper reframes finite superrotations as self-maps of Minkowski space that may be multivalued or non-surjective, thereby removing bulk defects but inducing defects on the celestial sphere. It derives explicit finite-superrotation transformations in Bondi and Newman-Unti gauges, showing how the transformed metric is the pullback of Minkowski space and introducing the principal domain to handle multivaluedness for cases like G(z)=z^α. Through detailed analysis of constant-(u,r) and constant-(u,ρ) surfaces, it reveals novel intrinsic geometries (cosmic footballs and horns) and clarifies how these surfaces intersect null infinity, with NU gauge offering a natural hyperbolic slicing perspective. The work discusses relaxing boundary conditions to accommodate a broader class of celestial-sphere metrics, the absence of bulk defects, and potential twistorial formulations, highlighting implications for the structure of asymptotic symmetries and flat-space holography.

Abstract

Superrotations arise from singular vector fields on the celestial sphere in asymptotically flat space, and their finite integrated versions have been argued by Strominger and Zhiboedov to insert cosmic strings into the spacetime. In this work, we argue for an alternative definition of the action of superrotations on Minkowski space that avoids introducing any defects. This involves realizing the finite superrotation not as a diffeomorphism between spaces, but as a mapping of Minkowski space to itself that may be multivalued or non-surjective. This eliminates any defects in the bulk spacetime at the expense of allowing for defects in the boundary celestial sphere metric. We further explore the geometry of the spatial surfaces in the superrotated spaces, and note that they intersect null infinity at the singularity of the superrotation, causing a breakdown in the large $r$ asymptotic expansion there. To determine how these surfaces embed into Minkowski space, a derivation of the finite superrotation transformation is presented in both Bondi and Newman-Unti gauges. The latter is particularly interesting, since the superrotations are shown to preserve the hyperbolic slicing of Minkowski space in Newman-Unti gauge, and this gauge also provides a means for extending the geometry beyond the Bondi coordinate patch. We argue that the new interpretation for the action of superrotations on spacetime motivates consideration of a wider class of celestial sphere metrics and asymptotic symmetry groups.

Figures (8)

• Figure 1: Isometric embedding of the cosmic football $(u=1,r=4,\alpha=1.4)$ into $\mathbb{R}^3$
• Figure 2: Isometric embedding of the cosmic horn $(u=-1,r=0.6,\alpha=0.8)$ into $\mathbb{R}^3$. The embedding breaks down shortly after it begins to flare out, becoming too negatively curved to embed into $\mathbb{R}^3$ in a way that preserves rotational symmetry.
• Figure 3: Bondi and Newman-Unti surfaces with $\alpha>1$, $u>0$. Since the surfaces are rotationally symmetric, the $\phi$ coordinate is suppressed, and the red and blue curves denote antipodal points on the surface at $\phi=0$ and $\phi=\frac{\pi}{\alpha}$.
• Figure 4: Bondi and Newman-Unti surfaces with $\alpha>1$, $u<0$.
• Figure 5: Bondi and Newman-Unti surfaces with $\alpha<1$, $u>0$.