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New conserved currents for vacuum space-times in dimension four with a Killing vector

A new family of conserved currents for vacuum space-times with a Killing vector is presented. The currents are constructed from the superenergy tensor of the Mars-Simon tensor and using the positivity properties of the former we find that the conserved charges associated to the currents have natural positivity properties in certain cases. Given the role played by the Mars-Simon tensor in local and semi-local characterisations of the Kerr solution, the currents presented in this work are useful to construct non-negative scalar quantities characterising Kerr initial data (known in the literature as non-Kerrness) which in addition are conserved charges.