Hidden symmetry of spacetime and Einstein metric

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05332
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Particle/Nuclear/Cosmic ray/Astro physics
• Research Institution: Setsunan University
• Principal Investigator: Yasui Yukinori 摂南大学, 理工学部, 教授 (30191117)
• Research Collaborator: HOURI TSUYOSHI
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: キリングテンソル場 / ブラックホール

Outline of Final Research Achievements

Killing tensor fields have been thought of as describing hidden symmetry of space(-time) since they are in one-to-one correspondence with polynomial first integrals of geodesic equations. Since many problems in classical mechanics can be formulated as geodesic problems in curved spaces and spacetimes, solving the defining equation for Killing tensor fields (the Killing equation) is a powerful way to integrate equations of motion. Thus it has been desirable to formulate the integrability conditions of the Killing equation, which serve to determine the number of linearly independent solutions and also to restrict the possible forms of solutions tightly. We show the prolongation for the Killing equation in a manner that uses Young symmetrizers. The prolonged equations can be viewed as the equations for parallel sections of the vector bundle, whose fibre is the space of differential forms. Using these equations, we provide the integrability conditions explicitly.

Free Research Field

数理物理学

Academic Significance and Societal Importance of the Research Achievements

キリング方程式の可積分条件を調べる研究は長い歴史を持つ。しかしながら，これまでに得られている結果は非常に複雑なものであり，大がかりなコンピュータ計算を行わない限り時空上にキリングテンソルが存在するかどうかを判定することは難しい。本研究では，キリング方程式を延長することにより，時空の曲率テンソルを使って可積分条件を陽に表すことに成功した。これにより，キリングテンソルの存在条件は，「ヤング図形から定まる曲率テンソル=0」という方程式と同値になる。この結果は，キリング方程式の可積分条件を簡単かつ明瞭に表現しており，多くの力学系への応用が期待できる。