{"title":"具有非各向同性质量的时空的辐射和渐近性","authors":"Lydia Bieri","doi":"10.4310/pamq.2024.v20.n4.a4","DOIUrl":null,"url":null,"abstract":"We derive new results on radiation, angular momentum at future null infinity and peeling for a general class of spacetimes. For asymptotically-flat solutions of the Einstein vacuum equations with a term homogeneous of degree $-1$ in the initial data metric, that is it may include a non-isotropic mass term, we prove new detailed behavior of the radiation field and curvature components at future null infinity. In particular, the limit along the null hypersurface $C_u$ as $t \\to \\infty$ of the curvature component $\\rho =\\frac{1}{4}{R_{3434}}$ multiplied with $r^3$ tends to a function $P(u, \\theta, \\phi)$ on $\\mathbb{R} \\times S^2$. When taking the limit $u \\rightarrow + \\infty$ (which corresponds to the limit at spacelike infinity), this function tends to a function $P^+(\\theta, \\phi)$ on $S^2$. We prove that the latter limit does not have any $l=1$ modes. However, it has all the other modes, $l = 0, l \\geq 2$. Important derivatives of crucial curvature components do not decay in $u$, which is a special feature of these more general spacetimes We show that peeling of the Weyl curvature components at future null infinity stops at the order $r^{-3}$, that is $(r^{-4}|u|^{+1}$, for large data, and at order $r^{-\\frac{7}{2}}$ for small data. Despite this fact, we prove that angular momentum at future null infinity is well defined for these spacetimes, due to the good behavior of the $l=1$ modes involved.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Radiation and Asymptotics for Spacetimes with Non-Isotropic Mass\",\"authors\":\"Lydia Bieri\",\"doi\":\"10.4310/pamq.2024.v20.n4.a4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We derive new results on radiation, angular momentum at future null infinity and peeling for a general class of spacetimes. For asymptotically-flat solutions of the Einstein vacuum equations with a term homogeneous of degree $-1$ in the initial data metric, that is it may include a non-isotropic mass term, we prove new detailed behavior of the radiation field and curvature components at future null infinity. In particular, the limit along the null hypersurface $C_u$ as $t \\\\to \\\\infty$ of the curvature component $\\\\rho =\\\\frac{1}{4}{R_{3434}}$ multiplied with $r^3$ tends to a function $P(u, \\\\theta, \\\\phi)$ on $\\\\mathbb{R} \\\\times S^2$. When taking the limit $u \\\\rightarrow + \\\\infty$ (which corresponds to the limit at spacelike infinity), this function tends to a function $P^+(\\\\theta, \\\\phi)$ on $S^2$. We prove that the latter limit does not have any $l=1$ modes. However, it has all the other modes, $l = 0, l \\\\geq 2$. Important derivatives of crucial curvature components do not decay in $u$, which is a special feature of these more general spacetimes We show that peeling of the Weyl curvature components at future null infinity stops at the order $r^{-3}$, that is $(r^{-4}|u|^{+1}$, for large data, and at order $r^{-\\\\frac{7}{2}}$ for small data. Despite this fact, we prove that angular momentum at future null infinity is well defined for these spacetimes, due to the good behavior of the $l=1$ modes involved.\",\"PeriodicalId\":54526,\"journal\":{\"name\":\"Pure and Applied Mathematics Quarterly\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-07-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Pure and Applied Mathematics Quarterly\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/pamq.2024.v20.n4.a4\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Pure and Applied Mathematics Quarterly","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/pamq.2024.v20.n4.a4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Radiation and Asymptotics for Spacetimes with Non-Isotropic Mass
We derive new results on radiation, angular momentum at future null infinity and peeling for a general class of spacetimes. For asymptotically-flat solutions of the Einstein vacuum equations with a term homogeneous of degree $-1$ in the initial data metric, that is it may include a non-isotropic mass term, we prove new detailed behavior of the radiation field and curvature components at future null infinity. In particular, the limit along the null hypersurface $C_u$ as $t \to \infty$ of the curvature component $\rho =\frac{1}{4}{R_{3434}}$ multiplied with $r^3$ tends to a function $P(u, \theta, \phi)$ on $\mathbb{R} \times S^2$. When taking the limit $u \rightarrow + \infty$ (which corresponds to the limit at spacelike infinity), this function tends to a function $P^+(\theta, \phi)$ on $S^2$. We prove that the latter limit does not have any $l=1$ modes. However, it has all the other modes, $l = 0, l \geq 2$. Important derivatives of crucial curvature components do not decay in $u$, which is a special feature of these more general spacetimes We show that peeling of the Weyl curvature components at future null infinity stops at the order $r^{-3}$, that is $(r^{-4}|u|^{+1}$, for large data, and at order $r^{-\frac{7}{2}}$ for small data. Despite this fact, we prove that angular momentum at future null infinity is well defined for these spacetimes, due to the good behavior of the $l=1$ modes involved.
期刊介绍:
Publishes high-quality, original papers on all fields of mathematics. To facilitate fruitful interchanges between mathematicians from different regions and specialties, and to effectively disseminate new breakthroughs in mathematics, the journal welcomes well-written submissions from all significant areas of mathematics. The editors are committed to promoting the highest quality of mathematical scholarship.