Dynamics of light cone cuts of null infinity

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Abstract
In this work we explore further consequences of a recently developed alternate formulation of general relativity, where the metric variable is replaced by families of surfaces as the primary geometric object of the theory—the (conformal) metric is derived from the surfaces—and a conformal factor that converts the conformal metric into an Einstein metric. The surfaces turn out to be characteristic surfaces of this metric. The earlier versions of the equations for these surfaces and conformal factor were local and included all vacuum metrics (with or without a cosmological constant). In this work, after first reviewing the basic theory, we specialize our study to spacetimes that are asymptotically flat. In this case our equations become considerably simpler to work with and the meaning of the variables becomes much more transparent. Several related insights into asymptotically flat spaces have resulted from this. (1) We have shown (both perturbatively and nonperturbatively for spacetimes close to Minkowski space) how a “natural” choice of canonical coordinates can be made that becomes the standard Cartesian coordinates of Minkowski space in the flat limit. (2) Using these canonical coordinates we show how a simple (completely gauge-fixed) perturbation theory off flat space can be formulated. (3) Using the rigid structure of the spacetime null cones (with their intersection with future null infinity) we show how the asymptotic symmetries (the BMS group or rather its Poincaré subgroup) can be extended to act on the interior of the spacetimes. This apparently allows us to define approximate Killing vectors and approximate symmetries. We also appear to be able to define a local energy-momentum vector field that is closely related to the asymptotic Bondi energy-momentum four-vector.

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