{"title":"(2+2) 哈密顿还原中的恩斯特方程的精确解","authors":"Jong Hyuk Yoon, Yeongji Kim, Seung Hun Oh","doi":"10.1007/s40042-024-01060-4","DOIUrl":null,"url":null,"abstract":"<div><p>We apply the method of Hamiltonian reduction without isometry as a way to find exact solutions to Einstein’s equations. To find exact solutions, we introduce two spatial Killing vector fields to the Einstein’s equations obtained through the Hamiltonian reduction, and derive the Ernst-like equation in the privileged coordinates. By solving the Ernst-like equation, we found a four-parameter family of exact solutions, one of which is interpreted as a deformation of the general Kasner spacetime. We extend our method to spacetimes where two independent gravitational degrees of freedom co-exist and interact with each other, and obtain a set of two partial differential equations satisfied by them. If we substitute a pre-fixed diagonal mode into these equations, and then the equations reduce to a single non-linear partial differential equation, which is interpreted as the equation of non-diagonal mode of gravitational waves propagating on the “background” spacetime determined by the diagonal mode. We choose three simplest “background” spacetimes, and discuss the corresponding non-diagonal modes in each case.</p></div>","PeriodicalId":677,"journal":{"name":"Journal of the Korean Physical Society","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Exact solutions to Ernst-like equation in (2+2) Hamiltonian reduction\",\"authors\":\"Jong Hyuk Yoon, Yeongji Kim, Seung Hun Oh\",\"doi\":\"10.1007/s40042-024-01060-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We apply the method of Hamiltonian reduction without isometry as a way to find exact solutions to Einstein’s equations. To find exact solutions, we introduce two spatial Killing vector fields to the Einstein’s equations obtained through the Hamiltonian reduction, and derive the Ernst-like equation in the privileged coordinates. By solving the Ernst-like equation, we found a four-parameter family of exact solutions, one of which is interpreted as a deformation of the general Kasner spacetime. We extend our method to spacetimes where two independent gravitational degrees of freedom co-exist and interact with each other, and obtain a set of two partial differential equations satisfied by them. If we substitute a pre-fixed diagonal mode into these equations, and then the equations reduce to a single non-linear partial differential equation, which is interpreted as the equation of non-diagonal mode of gravitational waves propagating on the “background” spacetime determined by the diagonal mode. We choose three simplest “background” spacetimes, and discuss the corresponding non-diagonal modes in each case.</p></div>\",\"PeriodicalId\":677,\"journal\":{\"name\":\"Journal of the Korean Physical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-05-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the Korean Physical Society\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40042-024-01060-4\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Korean Physical Society","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s40042-024-01060-4","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
Exact solutions to Ernst-like equation in (2+2) Hamiltonian reduction
We apply the method of Hamiltonian reduction without isometry as a way to find exact solutions to Einstein’s equations. To find exact solutions, we introduce two spatial Killing vector fields to the Einstein’s equations obtained through the Hamiltonian reduction, and derive the Ernst-like equation in the privileged coordinates. By solving the Ernst-like equation, we found a four-parameter family of exact solutions, one of which is interpreted as a deformation of the general Kasner spacetime. We extend our method to spacetimes where two independent gravitational degrees of freedom co-exist and interact with each other, and obtain a set of two partial differential equations satisfied by them. If we substitute a pre-fixed diagonal mode into these equations, and then the equations reduce to a single non-linear partial differential equation, which is interpreted as the equation of non-diagonal mode of gravitational waves propagating on the “background” spacetime determined by the diagonal mode. We choose three simplest “background” spacetimes, and discuss the corresponding non-diagonal modes in each case.
期刊介绍:
The Journal of the Korean Physical Society (JKPS) covers all fields of physics spanning from statistical physics and condensed matter physics to particle physics. The manuscript to be published in JKPS is required to hold the originality, significance, and recent completeness. The journal is composed of Full paper, Letters, and Brief sections. In addition, featured articles with outstanding results are selected by the Editorial board and introduced in the online version. For emphasis on aspect of international journal, several world-distinguished researchers join the Editorial board. High quality of papers may be express-published when it is recommended or requested.