How the non-metricity of the connection arises naturally in the classical theory of gravity

IF 1.2 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Bartłomiej Bąk, Jerzy Kijowski
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引用次数: 0

Abstract

Spacetime geometry is described by two–a priori independent–geometric structures: the symmetric connection Γ and the metric tensor g. Metricity condition of Γ (i.e. ∇g = 0) is implied by the Palatini variational principle, but only when the matter fields belong to an exceptional class. In case of a generic matter field, Palatini implies non-metricity of Γ. Traditionally, instead of the (first order) Palatini principle, we use in this case the (second order) Hilbert principle, assuming metricity condition a priori. Unfortunately, the resulting right-hand side of the Einstein equations does not coincide with the matter energy-momentum tensor. We propose to treat seriously the Palatini-implied non-metric connection. The conventional Einstein’s theory, rewritten in terms of this object, acquires a much simpler and universal structure. This approach opens a room for the description of the large scale effects in General Relativity (dark matter?, dark energy?), without resorting to purely phenomenological terms in the Lagrangian of gravitational field. All theories discussed in this paper belong to the standard General Relativity Theory, the only non-standard element being their (much simpler) mathematical formulation. As a mathematical bonus, we propose a new formalism in the calculus of variations, because in case of hyperbolic field theories the standard approach leads to nonsense conclusions.
连接的非度量性如何在经典引力理论中自然产生
时空几何由两个先验的独立几何结构描述:对称连接Γ 和度量张量 g。Γ 的度量性条件(即∇g = 0)隐含在帕拉蒂尼变分原理中,但只有当物质场属于一个特殊类别时才会出现。如果是一般物质场,帕拉蒂尼则意味着 Γ 的非度量性。传统上,在这种情况下,我们不使用(一阶)帕拉蒂尼原理,而是使用(二阶)希尔伯特原理,先验地假设度量条件。遗憾的是,由此得出的爱因斯坦方程的右侧与物质能动张量并不重合。我们建议认真对待帕拉蒂尼暗示的非度量联系。传统的爱因斯坦理论在这个对象的基础上重新书写后,获得了一个简单得多的通用结构。这种方法为描述广义相对论中的大尺度效应(暗物质、暗能量)开辟了空间,而无需诉诸引力场拉格朗日中的纯现象学术语。本文讨论的所有理论都属于标准广义相对论,唯一的非标准要素是它们的数学表述(简单得多)。作为数学奖励,我们在变分微积分中提出了一种新的形式主义,因为在双曲场理论中,标准方法会导致无意义的结论。
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来源期刊
Journal of Mathematical Physics
Journal of Mathematical Physics 物理-物理:数学物理
CiteScore
2.20
自引率
15.40%
发文量
396
审稿时长
4.3 months
期刊介绍: Since 1960, the Journal of Mathematical Physics (JMP) has published some of the best papers from outstanding mathematicians and physicists. JMP was the first journal in the field of mathematical physics and publishes research that connects the application of mathematics to problems in physics, as well as illustrates the development of mathematical methods for such applications and for the formulation of physical theories. The Journal of Mathematical Physics (JMP) features content in all areas of mathematical physics. Specifically, the articles focus on areas of research that illustrate the application of mathematics to problems in physics, the development of mathematical methods for such applications, and for the formulation of physical theories. The mathematics featured in the articles are written so that theoretical physicists can understand them. JMP also publishes review articles on mathematical subjects relevant to physics as well as special issues that combine manuscripts on a topic of current interest to the mathematical physics community. JMP welcomes original research of the highest quality in all active areas of mathematical physics, including the following: Partial Differential Equations Representation Theory and Algebraic Methods Many Body and Condensed Matter Physics Quantum Mechanics - General and Nonrelativistic Quantum Information and Computation Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory General Relativity and Gravitation Dynamical Systems Classical Mechanics and Classical Fields Fluids Statistical Physics Methods of Mathematical Physics.
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