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This paper establishes a mathematical proof of the blue-shift instability atthe sub-extremal Kerr Cauchy horizon for the linearised vacuum Einsteinequations. More precisely, we exhibit conditions on the $s=+2$ Teukolsky field,consisting of suitable integrated upper and lower bounds on the decay along theevent horizon, that ensure that the Teukolsky field, with respect to a framethat is regular at the Cauchy horizon, becomes singular. The conditions are inparticular satisfied by solutions of the Teukolsky equation arising fromgeneric and compactly supported initial data by the recent work [51] of Ma andZhang for slowly rotating Kerr.

Instability of the Kerr Cauchy horizon under linearised gravitational perturbations