On BMS Invariance of Gravitational Scattering (1312.2229v2)

Abstract: BMS+ transformations act nontrivially on outgoing gravitational scattering data while preserving intrinsic structure at future null infinity (I+). BMS- transformations similarly act on ingoing data at past null infinity (I-). In this paper we apply - within a suitable finite neighborhood of the Minkowski vacuum - results of Christodoulou and Klainerman to link I+ to I- and thereby identify "diagonal" elements BMS0 of (BMS+)X(BMS-). We argue that BMS0 is a nontrivial infinite-dimensional symmetry of both classical gravitational scattering and the quantum gravity S-matrix. It implies the conservation of net accumulated energy flux at every angle on the conformal S2 at I+. The associated Ward identity is shown to relate S-matrix elements with and without soft gravitons. Finally, BMS0 is recast as a U(1) Kac-Moody symmetry and an expression for the Kac-Moody current is given in terms of a certain soft graviton operator on the boundary of null infinity.

• The paper establishes that diagonal BMS symmetry (BMS0) generates an infinite-dimensional invariance for both classical and quantum gravitational scattering.
• It employs symmetry arguments and Ward identities to link soft graviton emissions with conservation laws, reinforcing Weinberg's soft graviton theorem.
• The study reformulates supertranslation invariance as a non-compact U(1) Kac-Moody symmetry, opening new avenues for understanding quantum gravity.

On BMS Invariance of Gravitational Scattering: A Summary

The paper "On BMS Invariance of Gravitational Scattering" by Andrew Strominger addresses the implications of the Bondi-Metzner-Sachs (BMS) group on gravitational scattering processes, both classically and quantum mechanically. The analysis is conducted within a framework that maintains the inherent structure at future null infinity, investigating how BMS transformations impact both outgoing and ingoing gravitational scattering data.

Overview of BMS Transformations

BMS transformations originate from the analysis of asymptotically flat spacetimes and constitute a set of "large" diffeomorphisms, which includes an infinite-dimensional subgroup of supertranslations. The supertranslation part allows for angle-dependent translations in retarded and advanced times, impacting the gravitational radiation at distinct moments and directions. Crucially, BMS symmetries can act separately on future null infinity () and past null infinity ($^-$), yielding BMS$^+$ and BMS$^-$ symmetry groups.

Diagonal BMS Elements and Classical Scattering

The paper centers around establishing links between the \ and $^-$ within a finite perturbative neighborhood of the Minkowski vacuum by utilizing insights from Christodoulou and Klainerman’s globally stable solutions of the Minkowski space. This leads to the identification of diagonal BMS elements, denoted as BMS$^0$, as symmetries that preserve the structure connecting both infinities.

Strominger argues that these diagonal symmetries imply an infinite-dimensional symmetry for classical gravitational scattering and the quantum gravity S-matrix. Notably, BMS$^0$ transformations conserve the accumulated net energy flux over each angle on the two-sphere at null infinity. This local conservation law is derived from the symmetries of the classical scattering solutions and further generates Ward identities in quantum theory.

Quantum Implications and Ward Identities

Quantum mechanically, the presence of BMS$^0$ invariance translates into symmetries of the S-matrix, which is postulated to relate in and out states within the Hilbert space. For gravitational scattering, the paper conjectures that, assuming the removal of infrared divergences is feasible, the S-matrix respects the BMS$^0$ symmetry.

The derived Ward identities establish connections between S-matrix elements with and without soft graviton insertions, a finding that is extended to reflect Weinberg's soft graviton theorem. These identities express conservation laws that link energy and infinitesimal supertranslations to soft gravitons, thereby recapturing familiar results from quantum field theory in gravity.

Supertranslations as Kac-Moody Symmetries

The paper further recasts the supertranslation invariance as a Kac-Moody symmetry on the two-sphere at null infinity, suggesting that supertranslation generators form a current algebra reminiscent of a non-compact $U(1)$ Kac-Moody algebra. This construction brings a fresh perspective to understanding gravitational symmetries at null boundaries, proposing a deeper symmetry structure potentially analogous to the well-established Kac-Moody framework in gauge theories.

Implications and Future Directions

The findings regarding BMS symmetries have significant implications for theoretical and mathematical formulations of quantum gravity. By drawing parallels with both classical and quantum scattering, the results emphasize the potential role of symmetries in resolving longstanding challenges such as infrared divergences and the conservation laws in gravitational processes. Moreover, the investigation opens avenues for exploring similar symmetry structures in related gauge theories, aligning with ongoing research in high-energy physics.

In conclusion, the paper provides a meticulous exploration of BMS symmetry in gravitational scattering, bridging classical results with contemporary quantum gravitational theories. The discussion of Kac-Moody type structures highlights novel aspects of asymptotic symmetry groups, suggesting their relevance in a broader context within theoretical physics. Future studies may aim to further substantiate these conjectures, especially concerning their concrete implementation in both classical and quantum gravitational regimes.