Mathematical Cosmology and Invariants in General Relativity

数学宇宙学和广义相对论中的不变量

• 批准号: RGPIN-2018-04045
• 负责人: Coley, Alan
• 金额: $ 1.46万
• 依托单位: Dalhousie University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712728
• 关键词: Mathematical; Cosmology; Invariants; General; Relativity

Summary: Mathematical Cosmology and Invariants in General Relativity In this research program, within mathematical cosmology we shall study inhomogeneous cosmological models using dynamical systems theory and computational methods. We shall study, using both exact solutions and numerics, the effect on the formation of large scale structure of inhomogeneous “spikes” that occur naturally and generically within general relativity and that generate small residual matter perturbations in the early universe. We shall also investigate the possible effects of spatial inhomogeneities, spatial curvature and non-Gaussianities in cosmology through backreaction effects. We propose novel approaches to averaging in cosmology using scalar invariants and within teleparallel gravity. We search for constraints on models of inflation inspired from string theory and higher-dimensional theories. We investigate models which re-collapse and then bounce into a new expansion phase, and determine whether black holes could persist through such a bounce. We have proven in four dimensions (4D) and in higher dimensions that generally a space- time is uniquely characterized by its scalar polynomial (curvature) invariants (SPI). We shall determine a complete (minimal) set of such SPI. We shall obtain a more practical way of determining the algebraic (Weyl) type of a higher dimensional spacetime employing SPI. We shall then algebraically classify higher dimensional black holes and seek new exact black hole solutions and study near horizon geometries. We shall obtain exact solutions in super-gravity and string theory that suffer no quantum corrections to all loop orders in 4D and in higher dimensions. We introduce the “foliation independent” concept of a geometric horizon, which is a surface distinguished by the vanishing of certain SPI (related to algebraical specialization), and motivate its use for the detection of the event horizon for stationary black holes in 4D. We propose to study the application of geometric horizons to more general dynamical black hole scenarios and in higher dimensions. This work is important not only for mathematical relativists, but hopefully also for numerical relativists and astrophysicists.

摘要：一般相对论中的数学宇宙学和不变性 在该研究计划中，在数学宇宙学中，我们将使用动态系统理论和计算方法研究不均匀的宇宙学模型。我们将使用精确的溶液和数字研究对自然和通常在一般相对性内发生的不均匀“尖峰”的大规模结构形成的影响，并在早期宇宙中产生小残留物质扰动。我们还将通过反应效应来研究空间不均匀性，空间曲率和非高斯的可能影响。我们提出了使用标量不变式和远程平行性重力内使用新颖的方法来平均宇宙学。我们搜索受弦理论和高维理论启发的通货膨胀模型的限制。我们研究了重新爆发然后反弹到新扩展阶段的模型，并确定黑洞是否可以通过这种反弹持续存在。 我们已经在四个维度（4D）中证明，在较高的维度上，通常时空的标量多项式（曲率）不变性（SPI）的特征是独特的。我们将确定此类SPI的完整（最小）集合。我们将获得一种更实用的方法来确定使用SPI的较高维度的代数（WEYL）类型。然后，我们将代数对更高维的黑洞进行分类，并寻求新的精确黑洞溶液并在地平线几何形状附近进行研究。我们将获得超级重力和弦理论的精确解决方案，这些解决方案对4D和较高维度的所有循环顺序均无量子校正。我们介绍了几何视野的“独立性”概念，该概念是由某些SPI消失（与代数专业化有关的）所区别的表面，并激励其用于检测事件范围4D固定黑洞的事件范围。我们建议研究几何视野在更通用的动态黑洞方案和更高维度中的应用。这项工作不仅对数学相对主义者很重要，而且希望对数字相对论家和天体物理学家来说也很重要。