Supertranslations call for superrotations (1102.4632v2)

Abstract: We review recent results on symmetries of asymptotically flat spacetimes at null infinity. In higher dimensions, the symmetry algebra realizes the Poincar\'e algebra. In three and four dimensions, besides the infinitesimal supertranslations that have been known since the sixties, the algebras are evenly balanced because there are also infinitesimal superrotations. We provide the classification of central extensions of the bms3 and bms4 algebras. Applications and consequences as well as directions for future work are briefly indicated.

• The paper demonstrates that in 3 and 4 dimensions, asymptotic symmetries extend to infinite-dimensional supertranslations and superrotations.
• It contrasts the finite Poincaré symmetry in higher dimensions with the richer, infinite-dimensional structures governing lower-dimensional spacetimes.
• The authors detail gauge fixation methods and central extensions, paving the way for advancements in quantum gravity research.

Analysis of "Supertranslations call for superrotations"

In the paper "Supertranslations call for superrotations" by Glenn Barnich and Cedric Troessaert, the authors examine the symmetries of asymptotically flat spacetimes at null infinity across various dimensions. This paper hinges on the concept of the Bondi-Metzner-Sachs (BMS) algebra, a construct used to describe these symmetries, particularly in three and four dimensions.

Asymptotic Symmetries in Higher Dimensions

In higher-dimensional spacetimes, the symmetry algebra manifests as the Poincaré algebra. This is derived by solving the Killing equation for the background metric to its leading order and reveals a symmetry consisting of the semi-direct sum of conformal Killing vectors of the $n-2$ sphere acting on the ideal of infinitesimal supertranslations. Notably, in dimensions higher than four, the algebra reduces to that of Poincaré due to the collapse of supertranslations into ordinary translations.

Specificities in Three and Four Dimensions

Contrastively, in three and four dimensions, the symmetry algebra is infinite-dimensional. Besides the well-established infinitesimal supertranslations, infinitesimal superrotations emerge as well. This aspect broadens the symmetry scope significantly from what was initially anticipated with the Poincaré algebra.

Detailed Symmetries and Gauge Fixations

The authors further distinguish between asymptotic and complete gauge fixations. The asymptotic approach treats symmetries as quotient algebras modulo trivial transformations, making it advantageous for confirming asymptotic flatness in specific solutions. However, the complete gauge fixation method stands out for trimming out symmetries tied to non-physical degrees of freedom.

Analysis of bms$_3$ and bms$_4$

An exhaustive analysis of bms$_3$ reveals its isomorphism with the Galilean conformal algebra (gca) in two dimensions. The paper addresses representations pertinent to gravitational contexts, notably those affiliated with a lowest eigenvalue for the Hamiltonian associated with $t_0$.

For bms$_4$, additional superrotations join supertranslations, further complicating the symmetry algebra's structure. The algebra divides regular conformal Killing vectors on the two-sphere into two Witt algebras, leading to a compelling narrative for potential extensions of this algebra. In this context, the work suggests pathways for rectifying angular momentum intricacies in general relativity.

Central Extensions and Practical Implications

The exploration of central extensions in both three and four dimensions reveals that while familiar central extensions apply within the typical scope (e.g., central extensions in Witt algebra copies), distinct properties appear relative to supertranslations. This factor complicates comprehensive representations yet offers rich avenues for investigation.

Future Directions

Future research directions include an intensive evaluation of physically significant representations of bms$_3$ and bms$_4$, alongside constructing surface charge algebras around the recently articulated supertranslations and superrotations. The realization of these algebras as Lie algebroid 2-cocycles rather than simpler Lie algebra structures marks a departure point for understanding their potential in quantum gravitational analysis. Such explorations could apply techniques from two-dimensional conformal field theory, aligning with S-matrix approaches in quantum four-dimensional general relativity.

This paper stands as a critical step towards a deeper understanding of asymptotic symmetries in gravitational theories, particularly in how they can inform and guide future inquiries into the underpinning structures of the universe within the field of theoretical physics.