{"title":"黑洞扰动的准正模展开:双曲Keldysh方法","authors":"Jérémy Besson, José Luis Jaramillo","doi":"10.1007/s10714-025-03438-6","DOIUrl":null,"url":null,"abstract":"<div><p>We study quasinormal mode expansions by adopting a Keldysh scheme for the spectral construction of asymptotic resonant expansions. Quasinormal modes are first cast in terms of a non-selfadjoint problem by adopting, in a black hole perturbation setting, a spacetime hyperboloidal approach. Then the Keldysh expansion of the resolvent, built on bi-orthogonal systems, provides a spectral version of Lax-Phillips expansions on scattering resonances. We clarify the role of scalar product structures in the Keldysh setting [1], that prove non-necessary to construct the resonant expansions (in particular the quasinormal mode time-series at null infinity), but are required to define the (constant) excitation coefficients in the bulk resonant expansion. We demonstrate the efficiency and accuracy of the Keldysh spectral approach to (non-selfadjoint) dynamics, even beyond its limits of validity, in particular recovering Schwarzschild black hole late power-law tails. We also study early dynamics by exploring i) the existence of an earliest time of validity of the resonant expansion and ii) the interplay between overtones extracted with the Keldysh scheme and regularity. Specifically, we address convergence aspects of the series and, on the other hand, we implement non-modal analysis tools, namely assessing <span>\\(H^p\\)</span>-Sobolev dynamical transient growths and constructing <span>\\(H^p\\)</span>-pseudospectra. Finally, we apply the Keldysh scheme to calculate “second-order” quasinormal modes and complement the qualitative study of overtone distribution by presenting the Weyl law for the counting of quasinormal modes in black holes with different (flat, De Sitter, anti-De Sitter) spacetime asymptotics.</p></div>","PeriodicalId":578,"journal":{"name":"General Relativity and Gravitation","volume":"57 7","pages":""},"PeriodicalIF":2.8000,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10714-025-03438-6.pdf","citationCount":"0","resultStr":"{\"title\":\"Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh’s approach\",\"authors\":\"Jérémy Besson, José Luis Jaramillo\",\"doi\":\"10.1007/s10714-025-03438-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We study quasinormal mode expansions by adopting a Keldysh scheme for the spectral construction of asymptotic resonant expansions. Quasinormal modes are first cast in terms of a non-selfadjoint problem by adopting, in a black hole perturbation setting, a spacetime hyperboloidal approach. Then the Keldysh expansion of the resolvent, built on bi-orthogonal systems, provides a spectral version of Lax-Phillips expansions on scattering resonances. We clarify the role of scalar product structures in the Keldysh setting [1], that prove non-necessary to construct the resonant expansions (in particular the quasinormal mode time-series at null infinity), but are required to define the (constant) excitation coefficients in the bulk resonant expansion. We demonstrate the efficiency and accuracy of the Keldysh spectral approach to (non-selfadjoint) dynamics, even beyond its limits of validity, in particular recovering Schwarzschild black hole late power-law tails. We also study early dynamics by exploring i) the existence of an earliest time of validity of the resonant expansion and ii) the interplay between overtones extracted with the Keldysh scheme and regularity. Specifically, we address convergence aspects of the series and, on the other hand, we implement non-modal analysis tools, namely assessing <span>\\\\(H^p\\\\)</span>-Sobolev dynamical transient growths and constructing <span>\\\\(H^p\\\\)</span>-pseudospectra. Finally, we apply the Keldysh scheme to calculate “second-order” quasinormal modes and complement the qualitative study of overtone distribution by presenting the Weyl law for the counting of quasinormal modes in black holes with different (flat, De Sitter, anti-De Sitter) spacetime asymptotics.</p></div>\",\"PeriodicalId\":578,\"journal\":{\"name\":\"General Relativity and Gravitation\",\"volume\":\"57 7\",\"pages\":\"\"},\"PeriodicalIF\":2.8000,\"publicationDate\":\"2025-07-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s10714-025-03438-6.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"General Relativity and Gravitation\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10714-025-03438-6\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"ASTRONOMY & ASTROPHYSICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"General Relativity and Gravitation","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s10714-025-03438-6","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ASTRONOMY & ASTROPHYSICS","Score":null,"Total":0}
Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh’s approach
We study quasinormal mode expansions by adopting a Keldysh scheme for the spectral construction of asymptotic resonant expansions. Quasinormal modes are first cast in terms of a non-selfadjoint problem by adopting, in a black hole perturbation setting, a spacetime hyperboloidal approach. Then the Keldysh expansion of the resolvent, built on bi-orthogonal systems, provides a spectral version of Lax-Phillips expansions on scattering resonances. We clarify the role of scalar product structures in the Keldysh setting [1], that prove non-necessary to construct the resonant expansions (in particular the quasinormal mode time-series at null infinity), but are required to define the (constant) excitation coefficients in the bulk resonant expansion. We demonstrate the efficiency and accuracy of the Keldysh spectral approach to (non-selfadjoint) dynamics, even beyond its limits of validity, in particular recovering Schwarzschild black hole late power-law tails. We also study early dynamics by exploring i) the existence of an earliest time of validity of the resonant expansion and ii) the interplay between overtones extracted with the Keldysh scheme and regularity. Specifically, we address convergence aspects of the series and, on the other hand, we implement non-modal analysis tools, namely assessing \(H^p\)-Sobolev dynamical transient growths and constructing \(H^p\)-pseudospectra. Finally, we apply the Keldysh scheme to calculate “second-order” quasinormal modes and complement the qualitative study of overtone distribution by presenting the Weyl law for the counting of quasinormal modes in black holes with different (flat, De Sitter, anti-De Sitter) spacetime asymptotics.
期刊介绍:
General Relativity and Gravitation is a journal devoted to all aspects of modern gravitational science, and published under the auspices of the International Society on General Relativity and Gravitation.
It welcomes in particular original articles on the following topics of current research:
Analytical general relativity, including its interface with geometrical analysis
Numerical relativity
Theoretical and observational cosmology
Relativistic astrophysics
Gravitational waves: data analysis, astrophysical sources and detector science
Extensions of general relativity
Supergravity
Gravitational aspects of string theory and its extensions
Quantum gravity: canonical approaches, in particular loop quantum gravity, and path integral approaches, in particular spin foams, Regge calculus and dynamical triangulations
Quantum field theory in curved spacetime
Non-commutative geometry and gravitation
Experimental gravity, in particular tests of general relativity
The journal publishes articles on all theoretical and experimental aspects of modern general relativity and gravitation, as well as book reviews and historical articles of special interest.