Mathematical Cosmology and Invariants in General Relativity

总结：数学宇宙学和广义相对论中的不变量在这个研究项目中，在数学宇宙学中，我们将使用动力系统理论和计算方法研究非均匀宇宙模型。我们将研究，使用精确解和数值计算，对形成大尺度结构的非均匀的“尖峰”，自然和一般发生在广义相对论，并产生小的残余物质扰动在早期宇宙的影响。我们也将探讨空间不均匀性，空间曲率和非高斯性在宇宙学中可能的影响，通过反作用效应。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telepareline重力。我们搜索的限制模型的通货膨胀的启发弦理论和高维理论。我们研究的模型，重新崩溃，然后反弹到一个新的膨胀phase，并确定是否黑洞可以坚持通过这样的bounce.We已经证明，在四维（4D）和更高的dimensions，一般的时空是唯一的特点，其标量多项式（曲率）不变量（SPI）。我们将确定这样的SPI的完整（最小）集合。我们将得到一个更实用的方法来确定代数（外尔）型的高维时空采用SPI。然后，我们将代数分类高维黑洞，并寻求新的精确黑洞的解决方案和研究近视界几何。我们将在超引力和弦理论中得到精确解，这些解在4D和更高维度中对所有回路阶数都没有量子修正。我们介绍了“叶状独立”的概念的几何视界，这是一个表面的区别消失的某些SPI（代数专业化），并激励其使用的事件视界的检测静止黑洞在4D。我们建议研究应用几何视界更一般的动态黑洞的情况下，在更高的维度。这项工作不仅对数学相对主义者很重要，而且希望对数值相对主义者和天体物理学家也很重要。

Summary: Mathematical Cosmology and Invariants in General RelativityIn 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.