Mathematical Cosmology and Invariants in General Relativity

摘要：数学宇宙学和广义相对论中的不变量******在这个研究项目中，我们将在数学宇宙学中使用动力系统理论和计算方法研究非齐次宇宙学模型。我们将使用精确解和数值来研究对大尺度非均匀“尖峰”结构形成的影响，这种结构在广义相对论中自然和一般地发生，并在早期宇宙中产生小的残余物质扰动。我们还将通过反作用力研究空间非均匀性、空间曲率和非高斯性在宇宙学中的可能影响。我们提出了新的方法来平均在宇宙学中使用标量不变量和在远平行重力。我们从弦理论和高维理论中寻找暴胀模型的约束条件。我们研究了重新坍缩然后反弹到一个新的膨胀阶段的模型，并确定黑洞是否可以在这样的反弹中持续存在。******我们已经证明了在四维（4D）和高维中，通常一个时空是由它的标量多项式（曲率）不变量（SPI）唯一表征的。我们将确定这种SPI的完整（最小）集。我们将得到一种更实用的方法来确定高维时空的代数（Weyl）类型。然后，我们将对高维黑洞进行代数分类，寻求新的精确黑洞解，并研究近视界几何。我们将在超引力和弦理论中得到精确的解，在四维和高维中所有环阶都不受量子修正。我们引入了几何视界的“叶分独立”概念，这是一个以某些SPI消失（与代数专门化相关）为特征的表面，并将其用于探测4D中静止黑洞的事件视界。我们建议研究几何视界在更一般的动态黑洞场景和更高维度上的应用。这项工作不仅对数学相对论主义者很重要，而且对数值相对论主义者和天体物理学家也很重要。

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.