An extension principle for Einstein-Maxwell-charged scalar field system with finite Doppler exponent

We prove a new extension principle for the spherically symmetric Einstein–Maxwell–charged scalar field system. Rather than imposing conditions for mass ratio $2m\mathrm{/}r$ as in Christodoulou’s small-mass-ratio criterion or as in An–Chen–Singh’s proof [1] with $2m\mathrm{/}r<3\mathrm{/}8$, here we work through a proposal by An-Tan [2] with finite Doppler exponent (blueshift). By conducting two successive blow-up analyses, followed by a stationary comparison and a rigidity argument, we establish the uniform boundness of ${\alpha }^{\mathrm{\prime }}$ with ${\alpha }^{\mathrm{\prime }}={\mathrm{\partial }}_v\alpha \mathrm{/}{\mathrm{\partial }}_{v}r$ and $\alpha ={\mathrm{\partial }}_v(r\phi )\mathrm{/}{\mathrm{\partial }}_{v}r$. This provides the desired extension principle in An-Tan [2] and completes the picture of weak cosmic censorship developed therein. As we were about to release our preprint, Cicortas communicated to us that together with his collaborators, they concomitantly used a different approach to obtain the corresponding extension principle with finite blueshift (Doppler exponent) for the uncharged Einstein-scalar field system. Hence, we decided to put the current paper as an appendix to An-Tan [2], thereby completing the proof of the weak cosmic censorship for the Einstein-Maxwell-charged scalar field system. At the same time, we are working with Cicortas and his collaborators to write a new preprint containing both approaches for the Einstein-scalar field system.