Mathematical challenges of General Relativity
Mathematical challenges of General Relativity
复制标题
广义相对论的数学挑战
DOI：
10.1142/9789812777386_0003
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发表时间：
2007
影响因子：
3.5
通讯作者：
S. Klainerman
中科院分区：
文献类型：
--
作者：
S. Klainerman
我们概述了广义相对论中的一些主要开放问题以及有关有界 L 曲率猜想的一些新结果。广义相对论与量子力学一起提供了现代物理学的概念框架，但与前者不同的是，广义相对论受到数学家的关注较少。然而，我认为，没有任何既定的物理理论具有更令人印象深刻的数学谱系或更丰富的数学基础。事实上，爱因斯坦在纯粹的理论尝试中发现了它，试图找到一种能够调和狭义相对论与牛顿引力的理论。从根本上来说，这种调和既需要黎曼几何的语言，又需要闵可夫斯基用洛伦兹度量的语言重新表述狭义相对论。狭义相对论本身诞生于另一项伟大的理论努力，旨在调和经典力学的伽利略不变性与麦克斯韦方程组的洛伦兹不变性。这两种物理理论本身都具有丰富的数学结构，并且已经并将继续与其他数学产生极其富有成效的相互作用。例如，可以说，微分形式理论和霍奇理论深受麦克斯韦电磁学理论的影响，而变分微积分和辛几何则是在解开经典力学的数学结构的长期且卓有成效的尝试中诞生的。
We give an overview of some of the main open problems in General Relativity as well as some new results concerning the bounded L curvature conjecture. Together with Quantum Mechanics, General Relativity provides the conceptual framework of Modern Physics yet, unlike the former, General Relativity has received somewhat less attention from mathematicians. There is, however, no established physical theory, I contend, which has a more impressive mathematical pedigree or a more fertile mathematical ground. Indeed recall that Einstein discovered it in a purely theoretical attempt to find a theory which could reconcile Special Relativity with Newtonian Gravity. The reconciliation required, in a fundamental way, both the language of Riemannian Geometry and the reformulation, by Minkowski, of Special Relativity in the language of a Lorentz metric. Special Relativity itself was born in another grand theoretical effort to reconcile the Galilean invariance of Classical Mechanics with the Lorentzian invariance of the Maxwell equations. Both these physical theories have rich mathematical structures in their own right and have had, and continue to have, an extremely fruitful interaction with the rest of mathematics. It suffices to say, for example, that the theory of differential forms and Hodge theory were greatly influenced by Maxwell’s theory of Electromagnetism while Calculus of Variations and Symplectic Geometry were born from a long and extremely fruitful attempt to unravel the mathematical structure of Classical Mechanics.