Asymptotic Symmetries in the Gauge Fixing Approach and the BMS Group

These notes are an introduction to asymptotic symmetries in gauge theories, with a focus on general relativity in four dimensions. We explain how to impose consistent sets of boundary conditions in the gauge fixing approach and how to derive the asymptotic symmetry parameters. The different procedures to obtain the associated charges are presented. As an illustration of these general concepts, the examples of four-dimensional general relativity in asymptotically (locally) (A)dS$_4$ and asymptotically flat spacetimes are covered. This enables us to discuss the different extensions of the Bondi-Metzner-Sachs-van der Burg (BMS) group and their relevance for holography, soft gravitons theorems, memory effects, and black hole information paradox. These notes are based on lectures given at the XV Modave Summer School in Mathematical Physics.Comment: 56 pages, 2 figures, published versio

The Λ\Lambda-BMS4_4 group of dS4_4 and new boundary conditions for AdS4_4

Using the dictionary between Bondi and Fefferman-Graham gauges, we identify the analogues of the Bondi news, Bondi mass and Bondi angular momentum aspects at the boundary of generic asymptotically locally (A)dS$_4$ spacetimes. We introduce the $\Lambda$-BMS$_4$ group as the residual symmetry group of the metric in Bondi gauge after boundary gauge fixing. This group consists of infinite-dimensional non-abelian supertranslations and superrotations and it reduces in the asymptotically flat limit to the extended BMS$_4$ group. Furthermore, we present new boundary conditions for asymptotically locally AdS$_4$ spacetimes which admit $\mathbb R$ times the group of area-preserving diffeomorphisms as the asymptotic symmetry group. The boundary conditions amount to fix 2 components of the holographic stress-tensor while allowing 2 components of the boundary metric to fluctuate. They correspond to a deformation of a holographic CFT$_3$ which is coupled to a fluctuating spatial metric of fixed area.Comment: 42 pages, 1 figure, attached Mathematica notebook, published versio

Conserved currents in the Cartan formulation of general relativity

We derive the expressions for the local, on-shell closed co-dimension 2 forms in the Cartan formulation of general relativity and explicitly show their equivalence to those of the metric formulation.Comment: 14 pages, Proceedings of the workshop "About various kinds of interactions" in honour of Philippe Spindel, 4 & 5 June 2015, Mons, Belgiu

On the Various Extensions of the BMS Group

The Bondi-Metzner-Sachs-van der Burg (BMS) group is the asymptotic symmetry group of radiating asymptotically flat spacetimes. It has recently received renewed interest in the context of the flat holography and the infrared structure of gravity. In this thesis, we investigate the consequences of considering extensions of the BMS group in four dimensions with superrotations. In particular, we apply the covariant phase space methods on a class of first order gauge theories that includes the Cartan formulation of general relativity and specify this analysis to gravity in asymptotically flat spacetime. Furthermore, we renormalize the symplectic structure at null infinity to obtain the generalized BMS charge algebra associated with smooth superrotations. We then study the vacuum structure of the gravitational field, which allows us to relate the so-called superboost transformations to the velocity kick/refraction memory effect. Afterward, we propose a new set of boundary conditions in asymptotically locally (A)dS spacetime that leads to a version of the BMS group in the presence of a non-vanishing cosmological constant, called the $\Lambda$-BMS asymptotic symmetry group. Using the holographic renormalization procedure and a diffeomorphism between Bondi and Fefferman-Graham gauges, we construct the phase space of $\Lambda$-BMS and show that it reduces to the one of the generalized BMS group in the flat limit.Comment: PhD Thesis, 204 pages, 2 figure

Charge Algebra in Al(A)dSn_n Spacetimes

The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in $n$ dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent $2$-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the $2$-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the $\Lambda$-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in $n$ dimensions.Comment: 48 pages, 1 figure, published versio

Nonlinear oscillations, transition to chaos and escape in the Duffing system with non-classical damping

We investigate the power of a ripping head in the process of concrete cutting. Using nonlinear embedding methods we study the corresponding time series obtained during the cutting process. The calculated maximal Lyapunov exponent indicates the exponential divergence typical for chaotic or stochastic systems. The recurrence plots technique has been used to get nonlinear process statistics for identification and description of nonlinear dynamics, lying behind the cutting process

Coadjoint representation of the BMS group on celestial Riemann surfaces

The coadjoint representation of the BMS group in four dimensions is constructed in a formulation that covers both the sphere and the punctured plane. The structure constants are worked out for different choices of bases. The conserved current algebra of non-radiative asymptotically flat spacetimes is explicitly interpreted in these terms.Comment: 45 pages Latex file, minimal cosmetic changes in version

Conserved currents in the Palatini formulation of general relativity

We derive the expressions for the local, on-shell closed co-dimension 2 forms in the Palatini formulation of general relativity and explicitly show their on-shell equivalence to those of the metric formulation. When compared to other first order formulations, two subtleties have to be addressed during the construction: off-shell non-metricity and the fact that the transformation of the connection under infinitesimal diffeomorphisms involves second order derivatives of the associated vector fields.Comment: 12 Page