A radiative phase space for generalised BMS symmetry [HBNI Th266]

Abstract

Symmetries play an important role in modern physics by providing a basis for many con- servation laws and enabling the classification of elementary particles. Typically, gravita- tional systems do not possess any global symmetries, and one instead tries to describe the symmetries of the metric far away from the sources. In coordinates suited to the problem, one computes the asymptotic form of the metric at large values of some “radial” parameter. ter. The asymptotic symmetry group is arrived at by solving for the transformations that preserve this asymptotic form of the metric. The case of interest to us is one where the leading-order terms from the radial expansion are chosen to agree with the metric in flat spacetime, and the resulting symmetry group is an infinite-dimensional group known as the Bondi-Metzner-Sachs (BMS) group [1, 2]. The past decade has seen renewed interest in asymptotic symmetries, owing in large part to Weinberg’s soft graviton theorem being recast as a conservation law for the charges associated with the BMS symmetries. Soft theorems are certain factorization properties enjoyed by terms from an asymptotic expansion of the S-matrix in the energy of some massless particle about zero. Given a novel soft theorem, one could now glean informa- tion about the charges of hitherto unknown asymptotic symmetries. This is how the gen- eralized BMS (gBMS) group came about, the conservation law arising out of its charges being equivalent to the Cachazo-Strominger soft graviton theorem [3, 4]. To demonstrate that the conserved charges derived from soft theorem are indeed those linked to an asymptotic symmetry, an asymptotic form of the metric was postulated, and the asymptotic symmetries were derived thereof [4]. A radiative phase space is then constructed within which the canonical transformations generated by the charges agree with the asymptotic symmetry transformations. Such a radiative phase space was realized recently in [5], and the subject matter of this thesis is the “inversion" of this symplectic structure to yield explicit expressions for the Poisson brackets. The null boundary of a 4-dimensional asymptotically flat spacetime, called null infinity, has the topology of R ⇥ S 2. The dynamical variables of the phase space are constructed out of the leading and subleading terms (from a radial expansion) of the angular part of the metric in a small neighborhood of null infinity. These are known as the celestial metric and the shear tensor, respectively. Our strategy is to replace certain composite variables formed out of the shear and celestial metric by auxiliary ones, which facili- states the straightforward computation of the Poisson brackets in this larger “kinematical" phase space. Working with such auxiliary variables poses a certain technical issue, whose resolution became apparent only after the problem was analysed in a context where the constraints are linearized. After this, we return to the original phase space, by reimposing ing relations involving these auxiliary variables, the shear tensor, and the celestial metric through a constraint analysis à la Dirac. Conventional depictions of the radiative phase space includes soft modes, which are thought of as living on the corners of null infinity. In keeping with this, we find that Poisson brackets of the shear tensor is augmented by distributional terms at the boundaries of null infinity. Among the novel Poisson brackets we find are those involving the celestial metric and the radiative data, and these constitute the main results of this thesis.