Linear Instability of the Reissner–Nordstro¨m Cauchy Horizon Eavan Gleeson 7 1 Department of Pure Mathematics 0 2 n and Mathematical Statistics a J 3 2 University of Cambridge ] c q - Trinity Hall r g [ 1 v 8 6 6 6 0 . 1 0 7 1 : v i X r a A dissertation submitted for the degree of Master of Science 2nd August, 2016 Acknowledgements First and foremost, I would like to thank my supervisor Jonathan Luk for his continuous encourage- ment, his limitless enthusiasm for the theory of General Relativity and for innumerable instructive and stimulating discussions over the course of my studies. I would also like to thank Mihalis Dafermos and Arick Shao for their very helpful comments and insights on the original version of this thesis. Moreover, I am extremely grateful to the Cambridge Centre for Analysis, the Department of Pure MathematicsandMathematicalStatistics, UniversityofCambridge, andtheRobertGardinerMemorial Fund for their financial support. Finally, I would like to express my gratitude to Daniel Fitzpatrick, to my family and to my friends for their ceaseless support during this time. 2 Abstract This work studies solutions of the scalar wave equation (cid:3) φ=0 g on a fixed subextremal Reissner–Nordström 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 W1,2 sense near the Cauchy horizon loc in the black hole interior, and it follows easily that they are also singular in the W1,p sense for p > 2. loc On the other hand, the work of Franzen shows that such solutions are non-singular near the Cauchy horizon in the W1,1 sense. Motivated by these results, we investigate the strength of the singularity at loc theCauchyhorizon. Weidentifyasufficientconditionontheblackholeinterior(whichholdsforthefull subextremal parameter range 0<|q| 2. Indeed, given a compact neighbourhood U of any point of the Cauchy horizon, we have Lp(U)⊆L2(U) when p>2. From this it follows that if generic W1,p blow up of solutions does not loc hold, then generic W1,2 blow up also fails, a contradiction with [16]. loc WethushaveW1,1 stabilityandW1,p instabilityofsolutionsforp≥2, butuncertaintyremainsover loc loc theprecisenatureoftheinstability. Forphysicalreasons,itisimportanttounderstandandquantifythe 6 strength of the singularity. Our first result in this direction is the following conditional theorem, which holds for the full subextremal range of parameters with non-vanishing charge, i.e. 0 < Q = |q| < 1, M and identifies a condition on the black hole interior which, if violated, ensures generic W1,p blow up of loc solutions for 1 r > 0 are e+1 r− + − the roots of r2 −2Mr+q2 = 0). Then for each 1 < p < 2, generic smooth and compactly supported initial data for (1) on Σ give rise to solutions that are not in W1,p in a neighbourhood of any point on 0 loc the future Cauchy horizon CH+. Remark. Recall that if p≥2, W1,p instability holds for the whole subextremal range of parameters with loc non-vanishing charge (0 < Q < 1). Combining this with our main theorem yields W1,p instability for loc √ every p>1 for the parameter range 2 e 1 such that solutions of the linear wave equation with smooth and compactly supported data on a Cauchy hypersurface are in W1,p near the Cauchy horizon. This conjecture is supported by [15], in which Hintz–Vasy showed that for all non-degenerate Reissner– Nordström–deSitterspacetimes, solutionsareinH1/2+β neartheCauchyhorizonforsomeβ >0.3 (See also[7],[11]and[18],whereSchwarzschild–deSitter,KerrandKerr–deSitterspacetimesareconsidered.) Thus the strength of the singularity is different for the asymptotically flat and cosmological cases. This is ultimately due to the differing decay properties of solutions in the black hole exterior regions, namely in Reissner–Nordström solutions decay polynomially in the exterior whereas in Reissner–Nordström–de Sitter solutions decay exponentially in the exterior. Aswithotherinstabilityresults,Theorems1.1and1.2aremotivatedbythestrongcosmiccensorship conjecture. This conjecture, perhaps the most fundamental open problem in mathematical general 2The reason for this parameter restriction isdue to the need topropagate L1-type estimates froma null hypersurface throughtheblackholeinteriortotheeventhorizon. SeeTheorem2.1fordetails. 3Since it was moreover shown that the solution is smooth in the directions tangental to the Cauchy horizon, one can thinkthatthesingularityisone-dimensionalandthatH1/2 scaleslikeW1,1 neartheCauchyhorizon. 8 relativity, is the subject of a huge body of literature. We will not discuss the conjecture itself here, but refer the reader to Section 1.4 of [16] (and the references within) where it is discussed in detail. Here, it sufficestosaythat,iftrue,thestrongcosmiccensorshipconjecturewouldimplythatsmallperturbations oftheReissner–Nordströmspacetimeforthenonlinear Einstein–Maxwellequationsgiverisetoasingular Cauchy horizon. The linear wave equation which we study is regarded as a “poor man’s linearisation” of the Einstein–Maxwell equations, and so the generic W1,p blow-up result presented in this work is loc evidence in favour of the strong cosmic censorship conjecture. Our instability result strengthens the W1,2 result of Luk–Oh, implying that the Cauchy horizon is more singular than was implied by [16]. loc Strategy of Proof We now turn our attention to the proofs of the two theorems above. As Theorem 1.1 follows almost immediatelyfromtheproofofTheorem1.2, inparticularfromidentifyinghowtheproofofTheorem1.2 √ may fail outside the parameter range 2 e

Linear Instability of the Reissner-Nordstr\"om Cauchy Horizon

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