Gravitomagnetism from Temporal Dimensional Reduction

Abstract

We reduce the Taub-NUT metric dimensionally to three spatial dimensions by treating time as an extra curled dimension, and end up with the 3-dimensional Einstein field equations plus a corresponding Maxwell type equations for a gravitomagnetic field, associated with the NUT charge, which also acts as a source for the Einstein field equations. In this approach, the Taub-NUT metric can be envisaged as a (1 + 3)-dimensional analogue of the (1 + 4)-dimensional metric of the Kaluza- Klein theory. Hence, in four dimensions, it unifies gravitation and gravitomagnetism, associated with the NUT charge, in the same footing that the Kaluza-Klein theory unifies gravitation and electromagnetism in five dimensions. In fact, in this way, gravity and gravitomagnetism, associated with the NUT charge, appear as two distinct fields that emerge from the temporal dimensional reduction. We also introduce a relation between the 4-dimensional gravitational constant and the NUT charge.