quantum field theory - Are gravitomagnetic monopoles hypothesized?

My understanding is that gravitomagnetism is essentially the same relativistic effect as magnetism. If so, why is it that I've heard so much about magnetic monopoles, but never gravitomagnetic monopoles? Aren't each just as theoretically well-motivated as the other? In a similar vein, one always hears that "there is no independent existence of an electric or magnetic field, only the electromagnetic field" because E and B fields transform into each other under Lorentz boosts. Shouldn't the same statement apply to gravity and gravitomagnetism?

Answer

Gravitational monopoles are forbidden by the positive mass theorem--- any configuration of GR has positive mass, and therefore is an ordinary "pole" in the analogy with electromagnetism. The analogy is not very good, because the energy is always positive, unlike charge.

The reason magnetic monopoles make sense in EM is because of electric-magnetic duality, a parity violating symmetry between the electric and magnetic charge in the free Maxwell equations that is obviously broken by the fact that sources don't have magnetic charge. The gravitational field has many magnetic field analogs, the field is the Levi-Civita connection, which decomposes into lots of different nonrelativistic things in different components, but none of these can appear without an ordinary gravitational field at long distances.

But there are ways of embedding electromagnetism into GR, as Kaluza Klein showed, and then there are special solutions of GR that can be interpreted as magnetic monopoles of the electromagnetic reduction. There is also a whole industry of finding self-dual solutions to GR, where the duality property can be thought of as analogous to electric magnetic duality, because it involves the epsilon tensor. Self dual gravitational fields, and the decomposition of the SO(3,1) holonomy in GR into two SU(2)'s are both extremely important and fascinating topics which contain many exact solutions, and the inspiration for loop quantum gravity.

If you would like some insight into self-dual solutions, this is a better question than asking for gravitational analogs to magnetic monopoles, because these analogs are not there.