Linear Instability of the Reissner-Nordström Cauchy Horizon
Linear Instability of the Reissner-Nordström Cauchy Horizon
复制标题
Reissner-Nordström 柯西视界的线性不稳定性
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Eavan Gleeson
中科院分区：
文献类型：
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作者：
Eavan Gleeson
This work studies solutions of the scalar wave equation [Box_gphi=0] on a fixed subextremal Reissner-Nordstr"{o}m spacetime with non-vanishing charge $q$ and mass $M$. In a recent paper, Luk and Oh established that generic smooth and compactly supported initial data on a Cauchy hypersurface lead to solutions which are singular in the $W^{1,2}_{loc}$ sense near the Cauchy horizon in the black hole interior, and it follows easily that they are also singular in the $W^{1,p}_{loc}$ sense for $p>2$. On the other hand, the work of Franzen shows that such solutions are non-singular near the Cauchy horizon in the $W^{1,1}_{loc}$ sense. Motivated by these results, we investigate the strength of the singularity at the Cauchy horizon. We identify a sufficient condition on the black hole interior (which holds for the full subextremal parameter range $0<|q|<M$) ensuring $W^{1,p}_{loc}$ blow up near the Cauchy horizon of solutions arising from generic smooth and compactly supported data for every $1<p<2$. We moreover prove that provided the spacetime parameters satisfy $frac{2sqrt e}{e+1}<frac{|q|}{M}<1$, we in fact have $W^{1,p}_{loc}$ blow up near the Cauchy horizon for such solutions for every $1<p<2$. This shows that the singularity is even stronger than was implied by the work of Luk and Oh for this restricted parameter range.
This work studies solutions of the scalar wave equation [Box_gphi=0] on a fixed subextremal Reissner-Nordstr"{o}m spacetime with non-vanishing charge $q$ and mass $M$. In a recent paper, Luk and Oh established that generic smooth and compactly supported initial data on a Cauchy hypersurface lead to solutions which are singular in the $W^{1,2}_{loc}$ sense near the Cauchy horizon in the black hole interior, and it follows easily that they are also singular in the $W^{1,p}_{loc}$ sense for $p>2$. On the other hand, the work of Franzen shows that such solutions are non-singular near the Cauchy horizon in the $W^{1,1}_{loc}$ sense. Motivated by these results, we investigate the strength of the singularity at the Cauchy horizon. We identify a sufficient condition on the black hole interior (which holds for the full subextremal parameter range $0<|q|<M$) ensuring $W^{1,p}_{loc}$ blow up near the Cauchy horizon of solutions arising from generic smooth and compactly supported data for every $1<p<2$. We moreover prove that provided the spacetime parameters satisfy $frac{2sqrt e}{e+1}<frac{|q|}{M}<1$, we in fact have $W^{1,p}_{loc}$ blow up near the Cauchy horizon for such solutions for every $1<p<2$. This shows that the singularity is even stronger than was implied by the work of Luk and Oh for this restricted parameter range.