Killing Reduction of 5-Dimensional Spacetimes

Abstract

In a 5-dimensional spacetime ($M,g_{ab}$) with a Killing vector field ${\xi }^a$ which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of ${\xi }^a$ gives a 4-dimensional space $S$. The reduction of ($M,g_{ab}$) is studied in the geometric language, which is a generalization of Geroch's method for the reduction of 4-dimensional spacetime. A 4-dimensional gravity coupled to a vector field and a scalar field on $S$ is obtained by the reduction of vacuum Einstein's equations on $M$, which gives also an alternative description of the 5-dimensional Kaluza-Klein theory. Besides the symmetry-reduced action from the Hilbert action on $M$, an alternative action of the fields on $S$ is also obtained, the variations of which lead to the same fields equations as those reduced from the vacuum Einstein equation on $M$.

9 pages, references added

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Cited by in corpus (8)

• 5-dimensional Brans-Dicke Theory and Cosmic Acceleration
• Five-dimensional generalized $f(R)$ gravity with curvature-matter coupling
• Five-dimensional PPN formalism and experimental test of Kaluza-Klein theory
• Five-dimensional metric $f(R)$ gravity and the accelerated universe
• Palatini Formalism of 5-Dimensional Kaluza-Klein Theory
• Riemann Tensor of the Ambient Universe, the Dilaton and the Newton's Constant
• Connection Dynamics of Reduced 5-dimensional Kaluza-Klein Theory and Its Deparametrization
• State-Relevant Maxwell's Equation from Kaluza-Klein Theory