Shock Waves and Geometry

冲击波和几何

基本信息

  • 批准号:
    9802473
  • 负责人:
  • 金额:
    $ 11.7万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    1998
  • 资助国家:
    美国
  • 起止时间:
    1998-07-15 至 2002-06-30
  • 项目状态:
    已结题

项目摘要

The research under this award investigates connections between geometry and the mathematical theory of shock waves. This includes a continuation of the authors' recent work on developing a mathematical theory of shock-waves that applies to the Einstein equations - the equations that describe the time evolution of spacetime curvature, the gravitational field according to general relativity. Shock waves are relevent here because the compressible Euler equations appear as a subsystem of the Einstein equations, and emerge exactly in the limit of low velocities and weak gravitational fields. The work includes the construction of a new cosmological model in which the expansion of the universe, (as measured by the Hubble constant), is modeled as the effect of a great explosion that generates a shock wave at the leading edge, something like a nuclear explosion into a static background, except on an enormous scale, instead of by the standard cosmological model in which it is assumed that the entire universe is expanding at a rate measured by the Hubble law. If successful, the results will provide the first and simplest model for cosmology under the assumption that the Hubble constant measures only a localized expansion, and consistent with the energy density and background radiation levels observed in the universe today. This research also has an interesting philosophical implication in that, if there were a shock wave present from the Big Bang, then, in contrast to the standard model of cosmology, there would be an associated loss of information that would make it impossible to reconstruct the details of the initial event from present data. In addition, a covariant version of Glimm's Method will be developed in an attempt to provide the first general existence theory of shock wave interactions for the Einstein equations. This is a step toward developing principles that can be applied to the numerical simulation of shock waves in general relativity. These studies will also have implications for classical fluids because the Einstein equations provide a unique setting in which natural geometrical constructs put a handle on the fluid dynamics. This work dovetails with ongoing research into how the large time behavior of shock wave solutions is related to the geometry of the Lie Algebra that gives the scattering picture forshock wave interactions. Thus this research connects the theory of shock waves and geometry on several levels. Einstein's equations of general relativity describe the large scale developmentof the universe. Although they were first proposed more than eighty years ago,they still pose formidable mathematical challenges. This research will continue earlier work in which exact shock wave solutions of these equations were found.A mathematical theory will be developed that leads to alternatives of the standard"Big Bang" theory and that allows a general theoretical treatment of shock waves within the general theory of relativity. Since these equations contain the classical fluid dynamics equations as a special case, the research will also lead to new insights for these more familiar problems.
该奖项下的研究调查几何和冲击波的数学理论之间的联系。 这包括作者最近关于发展适用于爱因斯坦方程的冲击波数学理论的工作的延续-爱因斯坦方程描述时空曲率的时间演化,根据广义相对论的引力场。 激波在这里是相关的,因为可压缩欧拉方程出现作为爱因斯坦方程的一个子系统,并出现在低速度和弱引力场的限制。 这项工作包括构建一个新的宇宙学模型,其中宇宙的膨胀,(由哈勃常数测量),被建模为在前缘产生冲击波的大爆炸的效果,类似于静态背景中的核爆炸,除了在巨大的规模上,而不是标准的宇宙学模型,其中假设整个宇宙正在以哈勃定律测量的速度膨胀。 如果成功的话,这些结果将提供第一个也是最简单的宇宙学模型,假设哈勃常数只测量局部膨胀,并且与今天在宇宙中观察到的能量密度和背景辐射水平一致。这项研究还有一个有趣的哲学含义,即如果存在来自大爆炸的冲击波,那么与宇宙学的标准模型相反,将存在相关的信息损失,这将使我们无法从现有数据中重建初始事件的细节。 此外,还将发展一种协变的格里姆方法,试图为爱因斯坦方程提供第一个普遍存在的激波相互作用理论。 这是朝着发展可应用于广义相对论中冲击波数值模拟的原理迈出的一步。 这些研究也将对经典流体产生影响,因为爱因斯坦方程提供了一个独特的设置,其中自然几何结构处理了流体动力学。 这项工作与正在进行的研究,如何大的时间行为的冲击波解决方案是相关的几何形状的李代数,给出了散射图片forshock波相互作用。因此,本研究在几个层次上将冲击波理论和几何学联系起来。爱因斯坦的广义相对论方程描述了宇宙的大尺度发展。 虽然它们在80多年前就被首次提出,但它们仍然构成了巨大的数学挑战。 这项研究将继续早期的工作,在这些方程的精确冲击波的解决方案,将开发一个数学理论,导致标准的“大爆炸”理论的替代品,并允许在广义相对论的冲击波的一般理论处理。 由于这些方程包含经典流体动力学方程作为特例,研究也将为这些更熟悉的问题带来新的见解。

项目成果

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John Temple其他文献

Resident duty hours around the globe: where are we now?
  • DOI:
    10.1186/1472-6920-14-s1-s8
  • 发表时间:
    2014-12-11
  • 期刊:
  • 影响因子:
    3.200
  • 作者:
    John Temple
  • 通讯作者:
    John Temple

John Temple的其他文献

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{{ truncateString('John Temple', 18)}}的其他基金

SHOCK-FREE AND SHOCK-WAVE DYNAMCS in GENERAL RELATIVITY and CLASSICAL FLUIDS
广义相对论和经典流体中的无冲击和冲击波动力学
  • 批准号:
    0707532
  • 财政年份:
    2007
  • 资助金额:
    $ 11.7万
  • 项目类别:
    Continuing Grant
Strong Shock Waves in Cosmology, General Relativity, and Classical Fluids
宇宙学、广义相对论和经典流体中的强冲击波
  • 批准号:
    0406096
  • 财政年份:
    2004
  • 资助金额:
    $ 11.7万
  • 项目类别:
    Standard Grant
Shock-Waves and Geometry
冲击波和几何
  • 批准号:
    0102493
  • 财政年份:
    2001
  • 资助金额:
    $ 11.7万
  • 项目类别:
    Continuing Grant
The Geometry of Shock Waves
冲击波的几何形状
  • 批准号:
    9500694
  • 财政年份:
    1995
  • 资助金额:
    $ 11.7万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Shock Waves and Conservation Laws
数学科学:冲击波和守恒定律
  • 批准号:
    9206631
  • 财政年份:
    1992
  • 资助金额:
    $ 11.7万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Shock Waves and Conservation Laws
数学科学:冲击波和守恒定律
  • 批准号:
    9006005
  • 财政年份:
    1990
  • 资助金额:
    $ 11.7万
  • 项目类别:
    Continuing Grant
U.S.-China Cooperative Research: Numerical and Analytical Studies in Nonlinear Waves
中美合作研究:非线性波的数值与分析研究
  • 批准号:
    8911134
  • 财政年份:
    1989
  • 资助金额:
    $ 11.7万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Systems of Conservation Laws: Analysis and Computation
数学科学:守恒定律系统:分析与计算
  • 批准号:
    8613450
  • 财政年份:
    1987
  • 资助金额:
    $ 11.7万
  • 项目类别:
    Continuing Grant
Mathematical Sciences Postdoctoral Research Fellowship
数学科学博士后研究奖学金
  • 批准号:
    8017157
  • 财政年份:
    1980
  • 资助金额:
    $ 11.7万
  • 项目类别:
    Fellowship Award

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  • 批准号:
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