The Riemann tensor and the Bianchi identity in 5D space-time

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• 作者：Mehran Taki ; ^{metaki@stu.yazd.ac.ir} ; Abolfazl Mirjalili ^{a.mirjalili@yazd.ac.ir}
• 关键词：Riemannian geometry ; Five-dimensional space&ndash ; time ; Bianchi identity ; Einstein's equation
• 刊名：Comptes Rendus Physique
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：18
• 期：1
• 页码：66-71
• 全文大小：245 K
• 卷排序：18

文摘

The initial assumption of theories with extra dimension is based on the efforts to yield a geometrical interpretation of the gravitation field. In this paper, using an infinitesimal parallel transportation of a vector, we generalize the obtained results in four dimensions to five-dimensional space–time. For this purpose, we first consider the effect of the geometrical structure of 4D space–time on a vector in a round trip of a closed path, which is basically quoted from chapter three of Ref. [5]. If the vector field is a gravitational field, then the required round trip will lead us to an equation which is dynamically governed by the Riemann tensor. We extend this idea to five-dimensional space–time and derive an improved version of Bianchi's identity. By doing tensor contraction on this identity, we obtain field equations in 5D space–time that are compatible with Einstein's field equations in 4D space–time. As an interesting result, we find that when one generalizes the results to 5D space–time, the new field equations imply a constraint on Ricci scalar equations, which might be containing a new physical insight.