Compatibility of the Dimensional Reduction and Variation Procedures for a Quadratic Curvature Model with a Kaluza-Klein Ansatz

TL;DR

This paper examines whether two different methods of deriving field equations in a higher-dimensional quadratic curvature model with a Kaluza-Klein ansatz produce consistent results after reduction to four dimensions.

Contribution

It provides a detailed analysis of the compatibility between variation and reduction procedures in deriving 4D field equations from a higher-dimensional quadratic curvature model.

Findings

The two procedures yield compatible field equations in the quadratic curvature Kaluza-Klein model.

The study clarifies the conditions under which the reduction and variation procedures are consistent.

Results support the validity of using either method for similar higher-dimensional theories.

Abstract

The introduction of extra dimensions is an invaluable strategy for the unification of gravity with other physical fields. Nevertheless, the matter in hand is to be eventually reduced to the actual 4D spacetime. The Kaluza-Klein theory is no exception to this well-known scheme. There are two procedures to obtain the field equations from a higher dimensional action. One can either take the variation of the effective action in that higher dimension and then reduce the resulting equations or reduce the higher dimensional action to the actual 4D and henceforward take the variations with respect to the constituent fields of the theory. Here, for the case of a quadratic curvature model with a Kaluza-Klein ansatz the field equations are obtained from the reduced action and compatibility of these two procedures is discussed in detail.