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Review and hystorical background:

Gravitomagnetism (GM), refers to a set of formal analogies between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. The most common version of GM is valid only far from isolated sources, and for slowly moving test particles. The GM equations coincide with equations which were first published in 1893, before general relativity, by Oliver Heaviside as a separate theory expanding Newton's law:

$\nabla \cdot \vec G = -4\pi\gamma\rho$

$\nabla \cdot \vec \Omega = 0$

$\nabla \times \vec G = - \dfrac{\partial \vec \Omega}{\partial t}$

$\nabla \times \vec \Omega = -\dfrac{4\pi\gamma}{c^2} \vec J + \dfrac{1}{c^2} \dfrac{\partial \vec G}{\partial t}$

$\vec G$ is gravitational field strength or gravitational acceleration, also called gravielectric for the sake of analogy; $\vec\Omega$ is intensity of torsion field or simply torsion, also called gravitomagnetic field; $\vec J $ is mass current density; $\gamma$ is gravitational constant.

Magnetic monopole and Maxwell's field equations:

It is known that the Maxwell's field equations have some asymmetry, in the absence of a magnetic monopole, although formally we can say that the problem can be solved theoretically (PAM Dirac and other works).

$\nabla \cdot \vec E = \dfrac{1}{\epsilon_0}\rho_e$

$\nabla \cdot \vec B = \mu_0 c \cdot g_m$, $g_m$ - magnetic monopole charge dencity.

$\nabla \times \vec E = \mu_0 J_{mag} - \dfrac{\partial \vec B}{\partial t}$, $J_{mag}$ - magnetic charge current

$\nabla \times \vec B = -\dfrac{1}{c^2 \epsilon_0} \vec J_{el} + \dfrac{1}{c^2} \dfrac{\partial \vec E}{\partial t}$, $J_{el}$ - electric charge current

General relativity and gravitomagnetic monopole:

Formally, a massive body in the linearized general relativity, is the gravielectric charge.

Now there is another interesting issue associated with the hypothesis of the existence gravimagnetic charge.

If we suppose its existence, what changes should be made to the equations of general relativity $G_{ik}= \kappa T_{ik}$ ($G_{ik}$ - Einstein tensor, $T_{ik}$ stress-energy tensor)? And what are the properties of such a charge?

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It looks like an exact duplicate to me. –  Ron Maimon May 24 '12 at 18:53
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There isn't any precise analogy between electromagnetism and gravity. In Newton's theory, only the gravitational acceleration (the gradient of the gravitational potential) exists at each point of space as an independent field; there is no independent extra "gravimagnetic" field.

In GR, the gravitational acceleration is given by $\Gamma^a_{bc}$, the Christoffel symbol, but it isn't antisymmetric in the same sense as $F_{\mu\nu}$ so one can't really Hodge-dualize it to get the dual magnetic field. One may speculate about the torsion, extra fields added to GR, but the observations exclude their existence at long distances at any significant couplings.

"Gravimagnetic" (literally) isn't an adjective used anywhere in serious physics literature. Instead, "gravitomagnetic" is sometimes used. But it refers to any effects - in GR or whatever right theory we consider - in which masses are moving and their impact is proportional to the velocity, much like Lorentz's force $v\times B$ in magnetism. See e.g. http://arxiv.org/abs/gr-qc/0207065 for a review. No full analogy with electromagnetism, of course.

Kaluza-Klein monopoles

The most interesting insight about "magnetic monopoles constructed purely from gravitational degrees of freedom" that one may discuss in serious physics are the so-called Kaluza-Klein monopoles. They appear in the Kaluza-Klein theory where the electromagnetic $U(1)$ gauge symmetry is geometrized as the group of rotations of an extra, circular coordinate of spacetime. In this setup, one may find the solutions of higher-dimensional Einstein's equations that looks like the Dirac magnetic monopole if the $g_{\mu 5}$ components of the metric are related to the electromagnetic potential $A_\mu$ according to the usual Kaluza-Klein dictionary. In 4+1D gravity compactified on circle, to imitate 3+1D gravity coupled to electromagnetism, Kaluza-Klein monopoles are point-like objects in 3+1D.

The KK monopoles play an important role in string/M-theory. In particular, D6-branes in type IIA string theory become KK monopoles (with 6 extra spatial dimensions in which the solution is extended/constant) if the coupling of type IIA string theory is sent to infinity to get M-theory in 11 dimensions. The M-theory (or 11D supergravity) solution for the KK monopole that becomes a D6-brane is actually completely non-singular and smooth.

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