Conservation Laws and Boundedness for Linearised Einstein--Maxwell Equations on the Reissner--Nordstr\"om Black Hole

We study the linearised Einstein--Maxwell equations on the Reissner--Nordstr\"om spacetime and derive the canonical energy conservation law in double null gauge. In the spirit of the work of Holzegel and the second author, we avoid any use of the hyperbolic nature of the Teukolsky equations and rely solely on the conservation law to establish control of energy fluxes for the gauge-invariant Teukolsky variables, previously identified by the third author, along all outgoing null hypersurfaces, for charge-to-mass ratio $\frac{|Q|}{M} < \frac{\sqrt{15}}{4}$. This yields uniform boundedness for the Teukolsky variables in Reissner--Nordstr\"om.