{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/273930"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/273930","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Instabilities in asymptotically AdS spacetimes","abstract":"In recent years, more and more efforts have been expended on the study of $n$-dimensional asymptotically anti-de Sitter spacetimes $(\\mathcal{M},g)$ as solutions to the Einstein vacuum equations \\begin{align*} \\mathrm{Ric}(g)=\\frac{2}{n-2}\\Lambda\\, g \\end{align*} with negative cosmological constant $\\Lambda$. This has been motivated mainly by the conjectured instability of these solutions. The author of this thesis joins these efforts with two contributions, which are themselves independent of each other. In the first part, we are concerned with a superradiant instability for $n=4$. For any cosmological constant $\\Lambda=-3/\\ell^2$ and any $\\alpha<9/4$, we find a Kerr-AdS spacetime $(\\mathcal{M},g_{\\mathrm{KAdS}})$, in which the Klein-Gordon equation \\begin{align*} \\Box_g\\psi+\\frac{\\alpha}{\\ell^2}\\psi=0 \\end{align*} has an exponentially growing mode solution satisfying a Dirichlet boundary condition at infinity. The spacetime violates the Hawking-Reall bound $r_+^2>|a|\\ell$. We obtain an analogous result for Neumann boundary conditions if $5/4<\\alpha<9/4$. Moreover, in the Dirichlet case, one can prove that, for any Kerr-AdS spacetime violating the Hawking-Reall bound, there exists an open family of masses $\\alpha$ such that the corresponding Klein-Gordon equation permits exponentially growing mode solutions. Our result provides the first rigorous construction of a superradiant instability for a negative cosmological constant. In the second part, we study perturbations of five-dimensional Eguchi-Hanson-AdS spacetimes exhibiting biaxial Bianchi IX symmetry. Within this symmetry class, the Einstein vacuum equations are equivalent to a system of non-linear partial differential equations for the radius $r$ of the spheres, the Hawking mass $m$ and $B$, a quantity measuring the squashing of the spheres, which satisfies a non-linear wave equation. First we prove that the system is well-posed as an initial-boundary value problem around infinity $\\mathcal{I}$ with $B$ satisfying a Dirichlet boundary condition. Second, we show that initial data in the biaxial Bianchi IX symmetry class around Eguchi-Hanson-AdS spacetimes cannot form horizons in the dynamical evolution.","abstract_html":"In recent years, more and more efforts have been expended on the study of $n$-dimensional asymptotically anti-de Sitter spacetimes $(\\mathcal{M},g)$ as solutions to the Einstein vacuum equations \\begin{align*} \\mathrm{Ric}(g)=\\frac{2}{n-2}\\Lambda\\, g \\end{align*} with negative cosmological constant $\\Lambda$. This has been motivated mainly by the conjectured instability of these solutions. The author of this thesis joins these efforts with two contributions, which are themselves independent of each other. In the first part, we are concerned with a superradiant instability for $n=4$. For any cosmological constant <span class=\"etd-inline-math\">\\Lambda=-3/\\ell<sup>2</sup></span> and any <span class=\"etd-inline-math\">&alpha;&lt;9/4</span>, we find a Kerr-AdS spacetime <span class=\"etd-inline-math\">(\\mathcal{M},g<sub><span class=\"etd-inline-math-roman\">KAdS</span></sub>)</span>, in which the Klein-Gordon equation \\begin{align*} \\Box_g\\psi+\\frac{\\alpha}{\\ell^2}\\psi=0 \\end{align*} has an exponentially growing mode solution satisfying a Dirichlet boundary condition at infinity. The spacetime violates the Hawking-Reall bound <span class=\"etd-inline-math\">r<sub>+</sub><sup>2</sup>&gt;|a|\\ell</span>. We obtain an analogous result for Neumann boundary conditions if <span class=\"etd-inline-math\">5/4&lt;&alpha;&lt;9/4</span>. Moreover, in the Dirichlet case, one can prove that, for any Kerr-AdS spacetime violating the Hawking-Reall bound, there exists an open family of masses <span class=\"etd-inline-math\">&alpha;</span> such that the corresponding Klein-Gordon equation permits exponentially growing mode solutions. Our result provides the first rigorous construction of a superradiant instability for a negative cosmological constant. In the second part, we study perturbations of five-dimensional Eguchi-Hanson-AdS spacetimes exhibiting biaxial Bianchi IX symmetry. Within this symmetry class, the Einstein vacuum equations are equivalent to a system of non-linear partial differential equations for the radius $r$ of the spheres, the Hawking mass $m$ and $B$, a quantity measuring the squashing of the spheres, which satisfies a non-linear wave equation. First we prove that the system is well-posed as an initial-boundary value problem around infinity $\\mathcal{I}$ with $B$ satisfying a Dirichlet boundary condition. Second, we show that initial data in the biaxial Bianchi IX symmetry class around Eguchi-Hanson-AdS spacetimes cannot form horizons in the dynamical evolution.","abstract_has_math":true,"creators":["Dold, Dominic Nicolas"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Dafermos, Mihalis"],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-04-28","date_published":"2018-04-28","updated_at":"2026-07-22T22:24:01Z","subjects":["mathematical general relativity","asymptotically locally AdS","Klein-Gordon equation","Einstein vacuum equations"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/43d087b7-757f-4e2b-94a4-5a9b20f43c5f/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000210847358"],"render_values":[{"text":"0000-0002-1084-7358","href":"https://orcid.org/0000-0002-1084-7358","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.21005","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Dafermos, Mihalis"]},{"key":"dc:creator","label":"Author","values":["Dold, Dominic Nicolas"]},{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000210847358"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2018-04-28"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/273930"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["mathematical general relativity","asymptotically locally AdS","Klein-Gordon equation","Einstein vacuum equations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/43d087b7-757f-4e2b-94a4-5a9b20f43c5f/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.21005"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9b170f4c-fe22-4ccd-85cf-23dc8e446f7d/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In recent years, more and more efforts have been expended on the study of $n$-dimensional asymptotically anti-de Sitter spacetimes $(\\mathcal{M},g)$ as solutions to the Einstein vacuum equations \\begin{align*} \\mathrm{Ric}(g)=\\frac{2}{n-2}\\Lambda\\, g \\end{align*} with negative cosmological constant $\\Lambda$. This has been motivated mainly by the conjectured instability of these solutions. The author of this thesis joins these efforts with two contributions, which are themselves independent of each other. In the first part, we are concerned with a superradiant instability for $n=4$. For any cosmological constant $\\Lambda=-3/\\ell^2$ and any $\\alpha<9/4$, we find a Kerr-AdS spacetime $(\\mathcal{M},g_{\\mathrm{KAdS}})$, in which the Klein-Gordon equation \\begin{align*} \\Box_g\\psi+\\frac{\\alpha}{\\ell^2}\\psi=0 \\end{align*} has an exponentially growing mode solution satisfying a Dirichlet boundary condition at infinity. The spacetime violates the Hawking-Reall bound $r_+^2>|a|\\ell$. We obtain an analogous result for Neumann boundary conditions if $5/4<\\alpha<9/4$. Moreover, in the Dirichlet case, one can prove that, for any Kerr-AdS spacetime violating the Hawking-Reall bound, there exists an open family of masses $\\alpha$ such that the corresponding Klein-Gordon equation permits exponentially growing mode solutions. Our result provides the first rigorous construction of a superradiant instability for a negative cosmological constant. In the second part, we study perturbations of five-dimensional Eguchi-Hanson-AdS spacetimes exhibiting biaxial Bianchi IX symmetry. Within this symmetry class, the Einstein vacuum equations are equivalent to a system of non-linear partial differential equations for the radius $r$ of the spheres, the Hawking mass $m$ and $B$, a quantity measuring the squashing of the spheres, which satisfies a non-linear wave equation. First we prove that the system is well-posed as an initial-boundary value problem around infinity $\\mathcal{I}$ with $B$ satisfying a Dirichlet boundary condition. Second, we show that initial data in the biaxial Bianchi IX symmetry class around Eguchi-Hanson-AdS spacetimes cannot form horizons in the dynamical evolution."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["87eda9de84448d1f82354d60eee3eb5f","9a593740c5af447bedaa02d2e6fd9829"]},{"key":"dc:title","label":"Title","values":["Instabilities in asymptotically AdS spacetimes"]}]}],"canonical_facts":{"dc:contributor.advisor":["Dafermos, Mihalis"],"dc:creator":["Dold, Dominic Nicolas"],"dc:creator.authoridentifier":["0000000210847358"],"dc:date.issued":["2018-04-28"],"dc:description.abstract":["In recent years, more and more efforts have been expended on the study of $n$-dimensional asymptotically anti-de Sitter spacetimes $(\\mathcal{M},g)$ as solutions to the Einstein vacuum equations \\begin{align*} \\mathrm{Ric}(g)=\\frac{2}{n-2}\\Lambda\\, g \\end{align*} with negative cosmological constant $\\Lambda$. This has been motivated mainly by the conjectured instability of these solutions. The author of this thesis joins these efforts with two contributions, which are themselves independent of each other. In the first part, we are concerned with a superradiant instability for $n=4$. For any cosmological constant $\\Lambda=-3/\\ell^2$ and any $\\alpha<9/4$, we find a Kerr-AdS spacetime $(\\mathcal{M},g_{\\mathrm{KAdS}})$, in which the Klein-Gordon equation \\begin{align*} \\Box_g\\psi+\\frac{\\alpha}{\\ell^2}\\psi=0 \\end{align*} has an exponentially growing mode solution satisfying a Dirichlet boundary condition at infinity. The spacetime violates the Hawking-Reall bound $r_+^2>|a|\\ell$. We obtain an analogous result for Neumann boundary conditions if $5/4<\\alpha<9/4$. Moreover, in the Dirichlet case, one can prove that, for any Kerr-AdS spacetime violating the Hawking-Reall bound, there exists an open family of masses $\\alpha$ such that the corresponding Klein-Gordon equation permits exponentially growing mode solutions. Our result provides the first rigorous construction of a superradiant instability for a negative cosmological constant. In the second part, we study perturbations of five-dimensional Eguchi-Hanson-AdS spacetimes exhibiting biaxial Bianchi IX symmetry. Within this symmetry class, the Einstein vacuum equations are equivalent to a system of non-linear partial differential equations for the radius $r$ of the spheres, the Hawking mass $m$ and $B$, a quantity measuring the squashing of the spheres, which satisfies a non-linear wave equation. First we prove that the system is well-posed as an initial-boundary value problem around infinity $\\mathcal{I}$ with $B$ satisfying a Dirichlet boundary condition. Second, we show that initial data in the biaxial Bianchi IX symmetry class around Eguchi-Hanson-AdS spacetimes cannot form horizons in the dynamical evolution."],"dc:format.checksum.md5":["87eda9de84448d1f82354d60eee3eb5f","9a593740c5af447bedaa02d2e6fd9829"],"dc:identifier.doi":["10.17863/CAM.21005"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9b170f4c-fe22-4ccd-85cf-23dc8e446f7d/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/273930"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/43d087b7-757f-4e2b-94a4-5a9b20f43c5f/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["mathematical general relativity","asymptotically locally AdS","Klein-Gordon equation","Einstein vacuum equations"],"dc:title":["Instabilities in asymptotically AdS spacetimes"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:01Z"}