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We demonstrate that the phase space of the soft sector of asymptotically flat gravity in four spacetime dimensions can be identified with that of a spherically symmetric finite casual diamond in Minkowski spacetime. The leading soft graviton mode is geometrically identified with the radial fluctuation of the causal diamond size, while the Goldstone mode involves both the radial fluctuation and its symplectic partner. This allows us to relate the radial fluctuations of the causal diamond with the asymptotic transverse fluctuations parametrized by the soft modes.
Ciambelli et al. (Wed,) studied this question.