Gravitomagnetism from Temporal Dimensional Reduction

TL;DR

This paper investigates a KK-inspired unification by treating time as an extra curled dimension in the Lorentzian Taub-NUT solution, aiming to unify gravitation with gravitomagnetism arising from the NUT charge. By performing a temporal dimensional reduction from $(1+3)$ to $(3)$ spatial dimensions, the authors derive a 3D Einstein sector coupled to a gravitomagnetic gauge field $\tilde{F}_{ab}$ and a dilaton $\Phi(r)$, governed by a system that includes $\tilde{G}_{ab} = -2 l^2 \Phi^2 \tilde{T}_{ab}^{(GM)} + (1/\Phi)[ \tilde{\nabla}_{a}(\partial_{b}\Phi) - \tilde{g}_{ab} \tilde{\Box} \Phi ]$, $\tilde{\nabla}^{a} \tilde{F}_{ab} = -3 (\partial^{a}\Phi)/\Phi \; \tilde{F}_{ab}$, and $\tilde{\Box} \Phi = - l^2 \Phi^3 \tilde{F}_{ab} \tilde{F}^{ab}$. A key result is the suggested relation $\tilde{\kappa}^2 \leftrightarrow 2l$ and $G \leftrightarrow (\oint dx^0 /(8\pi))\, l$, tying the 4D gravitational constant to the NUT charge and the time-circle size. The analysis shows that when time is not compactified, gravity and gravitomagnetism mix in 4D, whereas curling time yields a 3D theory with distinct fields and a dilaton mode, and regions where CTCs are removed via a signature change. The work opens avenues for Brans–Dicke/dilaton extensions and Euclidean/topological interpretations of the NUT charge.

Abstract

We propose that the Taub-NUT metric can be envisaged as a (3+1) dimensional analog of the Kaluza-Klein (4+1) dimensional metric. After dimensional reduction of the Taub-NUT metric to (3) spatial dimensions, by treating time as the extra curled dimension (since the closed time-like curves can exist in the Taub-NUT framework, such a dimensional reduction is justified), we end up with three dimensional Einstein field equations plus the Maxwell equations for the gravitomagnetic field, which also acts as a source to Einstein field equations. Hence, the Taub-NUT metric unifies gravity and the NUT charge related gavitomagnetism in four dimensions, at the same footing the Kaluza-Klein metric unifies gravity and electromagnetism in five dimensions. We also introduce a relation between the 4-dimensional gravitational constant and the NUT charge.