引用本文: 殷雅俊. 从空间到时间——张量的协变微分学及协变性思想的拓展1)[J]. 力学与实践, 2021, 43(6): 933-944.
YIN Yajun. FROM SPACE TO TIME - FURTHER DEVELOPMENT OF TENSOR COVARIANT DIFFERENTIAL THEORY AND COVARIANCE CONCEPT$^{1)}$[J]. MECHANICS IN ENGINEERING, 2021, 43(6): 933-944.
 Citation: YIN Yajun. FROM SPACE TO TIME - FURTHER DEVELOPMENT OF TENSOR COVARIANT DIFFERENTIAL THEORY AND COVARIANCE CONCEPT$^{1)}$[J]. MECHANICS IN ENGINEERING, 2021, 43(6): 933-944.

## FROM SPACE TO TIME - FURTHER DEVELOPMENT OF TENSOR COVARIANT DIFFERENTIAL THEORY AND COVARIANCE CONCEPT^1)

• 摘要: 本文聚焦于张量分析学中两类对称性破缺现象。一是基础性概念的对称性破缺：我们有张量对坐标的协变微分(简称空间协变微分),然而,没有明确定义过如下概念——张量对时间的协变微分(简称时间协变微分)。二是基本理论的对称性破缺：我们有空间域上张量的协变微分学,但没有时间域上张量的协变微分学。本文致力于弥补破缺的对称性。基于近年来的研究进展,回顾协变性思想的拓展历史,展示张量的时间协变微分概念的抽象过程,综述时间域上张量的协变微分学的构建历程,揭示空间域上的协变微分学与时间域上的协变微分学之间的对称性。

Abstract: This paper focuses on two types of symmetry breakages in tensor analysis. One is the symmetry breakage in the concept that there is covariant differential of tensor with respect to coordinate, but there is no one with respect to time. Another is the symmetry breakage in the theory that there is covariant differential theory of tensor in space field, but there is no one in time field. This paper tries to mend the symmetry breakages on the basis of recent years research progresses. Firstly the history of extending the covariance thought is reviewed. Then the process of abstracting the concept of covariant differential with respect to time is developed. Finally the procedure of constructing the covariant differential theory of tensor in time field is summarized. It ultimately reveals the symmetry between the covariant differential theory in time field and the covariant differential theory in space field.

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