Asymptotic Symmetries in the Gauge fixing Approach and the BMS group

Abstract

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 fourdimensional 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.

Surface charges toolkit for gravity

“…generalized Killing equation), 4) Compute the surface charge integral with those parameters (to get a differential charge), and 5) Integrate the differential charge on phase space. In the present notes we extensively explore the first 2 See for instance a recent review about the equivalence among the Iyer-Wald and Barnich-Brandt procedure [13]. Analogously, in [14] was shown that the of-shell Abbot-Deser-Tekin formalism is related to the Iyer-Wald and Barnich-Brandt-Compére formalism.…”

“…A further analysis allows for the obtained quantity to be interpreted as the entropy of the black hole. Here we stress that although the formula simplifies on the bifurcated horizon, one is also able to compute the same value for the entropy outside the horizon, on an arbitrary closed surface containing the black hole singularity, by using the full formula (3.25) 13 . The reason is that the full formula is protected by the conservation while the Noether potential is not.…”

“…A new result, as far as the authors' knowledge concerns, is about torsional gravity theories: We show two relevant examples where the charges are unaffected by the presence of non-dynamical torsion fields. 13 However, as in the standard case an extra input should be considered to fix the normalization of the horizon generator. The usual argument is based on a normalization of the Killing field at the asymptotic region.…”

The Λ-BMS4 charge algebra

“…The symplectic structure is defined in two steps. First, the presymplectic form is defined from covariant phase space methods [72][73][74] as we shortly review (see also [40,41]). Second, it is renormalized in accordance with the action [78].…”

“…In asymptotically flat Einstein gravity, leading [1][2][3], subleading/overleading [4][5][6][7][8][9][10] or higher subleading/overleading [11,12] relationships have been found in the infrared sector, which relate symmetries [5,6,9,[13][14][15][16][17][18][19][20][21], memory effects [7,12,[22][23][24][25][26][27][28][29][30][31][32][33][34][35] and scattering identities involving soft modes [36,37]. Partial reviews include [38][39][40][41]. In cosmology, such relationships have also been described [42][43][44]…”