I started studying the BMS symmetry in connection with the paper: http://arxiv.org/abs/1312.2229 and there are a few strange things I noticed.
First of all, from reading the original papers by Bondi, Metzner and Sachs, I know that the "BMS symmetry" is just an allowed subset of coordinate diffeomorphisms which leaves the asymptotic flatness of the space-time intact. However, when I read the paper above, the BMS symmetry is stated in form of a vector field eq. (2.10) and (2.14). Therefore, my first question:
How do I obtain from a given subset of coordinate diffeomorphisms (which is being considered a symmetry) a corresponding vector field a'la Killing vector?
Furthermore, later in the paper they go from the BMS vector fields to generators of BMS symmetry in eq. (3.3). They do not mention how to do it, so I'd like to know:
How do I go from vector fields characterizing a symmetry to an actual generator of the symmetry?
I realize that this is some pretty advanced stuff. I am grateful for any help or suggestion!
Answer
It is easiest to directly derive the form of the vector fields from the boundary conditions for asymptotically flat spacetimes. See for example this paper
http://arxiv.org/abs/1001.1541
or this one
http://arxiv.org/abs/1106.0213
Infinitesimally diffeomorphisms act on the metric via Lie derivative $\mathcal{L}_{\xi}g_{\mu \nu}$. In the case of BMS, you require this transformation to respect the boundary conditions on $g_{\mu \nu}$, for example $g_{uu}\approx -1 +\mathcal{O}(r^{-1})$. So the vector field should satisfy $\nabla_u \xi_u =\mathcal{O}(r^{-1})$. You may set up such equations for each component of the metric and its corresponding boundary conditions. In each case you require the vector field not to touch the "leading" Minkowski part of the metric. This is a system of differential equations, which you may then solve.
As far as the generators go, this is sort of addressed in the second paper I linked to. Though as far as eqn 3.3 goes, you may heuristically think of this just as the ADM hamiltonian weighted by a function on the sphere which turns the uniform time translation into a supertranslation (the spaces Andy is studying are Christodoulou-Klainerman spaces for which $i^0$ is a non-singular point and so you may hope to match $\mathscr I^+_-$ to $\mathscr I^-_+$ through $i^0$ where the ADM hamiltonian is defined). Demonstrating that these charges generate the correct transformations quantum mechanically is actually very subtle, as is discussed here
http://arxiv.org/abs/1401.7026
Hope this helps.
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