On the Behavior of Solutions of Einstein's Equations and Other Geometric Nonlinear Partial Differential Equations

关于爱因斯坦方程和其他几何非线性偏微分方程解的行为

• 批准号: 0099373
• 负责人: James Isenberg
• 金额: $ 11.1万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2004-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0099373&HistoricalAwards=false
• 关键词: Behavior; Solutions; Einstein; Equations; Other

Einstein's gravitational field theory (general relativity), besidesproviding the most accurate current model for the study of gravitationalphysics on the astrophysical and cosmological scales, is also the source ofa rich collection of mathematically interesting questions. This award willsupport a program of research which focusses on some of these, includingthe following:1) Study of the behavior of the gravitational field very close to the bigbang in cosmological solutions of Einstein's equations. Remarkably, itseems that even for solutions which appear to be very different some timeafter the big bang, the behavior close to the big bang is very similar: Itoscillates in a characteristic way. We can show this, numerically oranalytically, for simple families of solutions, and we hope to show it moregenerally.2) Construction of new sets of initial conditions for the gravitationalfield which juxtapose two known sets. This is not easy because the initialconditions must satisfy certain nonlinear partial differentialequations--the Einstein "constraints". We are developing techniques forsmoothly joining ( or "gluing") two solutions of the constraints into asingle solution.The first of these projects, although it ignores any quantum gravitationaleffects, could be very useful in our drive to understand the nature of thevery early universe. The second is of interest both mathematically andphysically. The mathematical interest stems from the fact that the Einsteinconstraint system is among the most complicated to which gluing studieshave been applied. The physical interest comes from the possiblility ofusing the developed techniques to set up initial data for black holecollisions and other astrophysical problems which are being intenselystudied numerically in anticipation of future data from LIGO.

爱因斯坦的引力场理论（总体相对论）除了提供了在天体物理和宇宙学量表上研究引力物质的最准确的当前模型，也是数学上有趣的问题的丰富收集。该奖项将支持一项研究计划，该计划的重点是其中一些，包括：1）在爱因斯坦方程的宇宙学解决方案中，对重力领域的行为非常接近。值得注意的是，即使对于似乎有很大不同的时间以前的大爆炸的解决方案，与大爆炸的行为也非常相似：以一种特征性的方式进行iToscill。对于简单的解决方案，我们可以用数字上的原始分析表明这一点，我们希望以概要表明它。2）构建重力田的新的初始条件，这并置了两个已知的集合。这并不容易，因为初始条件必须满足某些非线性部分差异化 - 爱因斯坦“约束”。我们正在开发一种技术，将约束的两种解决方案加入（或“粘合”）到Asingle解决方案中。尽管这些项目中的第一个忽略了任何量子重力，但在我们了解早期宇宙的性质方面可能非常有用。第二个是在数学上和物理上都令人感兴趣的。数学兴趣源于以下事实：Einsteinconstraint系统是应用胶水研究最复杂的事实。物理兴趣来自于开发技术为黑色全鼠和其他天体物理问题设置初始数据的可能性，这些数据是在数值上被深入研究的，这些数据是预期Ligo的未来数据。