New approaches to the study of gravitation in relativistic astrophysics

申请人与几名学生一起，在多年的时间里，开发了广泛使用的计算机代数系统GRTensor，这是应用微分几何领域的重要工具。我提出的研究计划以新的方式利用了这些专业知识。在这里，我将集中在这个程序的一个方面，曲率的梯度场。我们的工作涉及计算机代数和数值例程的新应用，目标是发展一种新的“曲率”观点——一种揭示以前无法获得的详细结构和见解的观点。给定任意不变量，总能构造相应的梯度场并研究其结果流。从本质上讲，我们已经把曲率的抽象概念变成了一种假设的流体，它的性质可以很容易地可视化。虽然这项研究还处于早期阶段，但我已经可以引用以下内容：(I)流区分不同类型的奇点。例如，当气流在克尔度规的赤道平面上寻找奇异环（真正的奇点）时，气流完全忽略了“壳聚焦奇点”，这证实了我多年前提出的观点，即这种奇点不是“引力”。（二）在非齐次宇宙学领域，人们错误地认为“里奇标量的达朗伯量”的散度表示“弱奇点”。这样的散度实际上是梯度流中的苛性，是没有后果的人工奇点。我们期望这种曲率可视化的新方法将演变成一种标准方法，对该领域的所有研究人员都有很大的好处。尽管还有很多工作要做，但总结这种方法的一种方法是注意到，当杀伤向量（对称方向）可用时，使用它们是明智的。从构造上讲，梯度流与杀戮流是正交的，并且在算法上总是可用的。应该指出的是，梯度流的使用并不局限于爱因斯坦的理论或四维空间。我的整体研究计划的这一方面为HQP的培训提供了特殊的机会，涉及两名研究生和一名暑期研究助理。

The applicant, along with several students and over a period of many years, developed the widely used computer algebra system GRTensor, an essential tool in the area of applied differential geometry. My proposed research program makes use of this expertise in new ways. Here I will concentrate on just one aspect of this program, gradient fields of curvature. Our work involves the novel use of both computer algebra and numerical routines with the objective being the development of a fresh view of "curvature" - a view that reveals detailed structure and insights previously unavailable. Given any invariant, one can always construct the associated gradient field and study the resultant flow. Essentially, we have turned the abstract idea of curvature into a hypothetical fluid the properties of which can be easily visualized. Although this research is in its early stages, I can already cite the following: (i) The flow distinguishes different types of singularities. For example, whereas the flow seeks out the singular ring in the equatorial plane of the Kerr metric (the real singularity), the flow completely ignores "shell focusing singularities", substantiating my claim, made many years ago, that such singularities are not "gravitational". (ii) In the area of inhomogeneous cosmology, it has been incorrectly claimed that the divergence of the "d'Alembertian of the Ricci scalar" signifies a "weak singularity". Such a divergence is actually a caustic in the gradient flow - and artificial singularity of no consequence. We expect that this fresh approach to the visualization of curvature will evolve into a standard approach much to the benefit to all researchers in the field. Whereas much work remains to be done, one way to summarize this approach is to note that when Killing vectors (directions of symmetry) are available, it is wise to make use of them. Gradient flows are, by construction, orthogonal to Killing flows and are always available - algorithmically. It should be noted that the use of gradient flows is not restricted to Einstein's theory nor to four dimensions. This aspect of my overall research program offers exceptional opportunities for the training of HQP and involves two graduate students and a summer research assistant.