Can anyone please clarify what one means by a vorticity in Space-Time? I'm aware of the vorticity in Fluid Mechanics. Is it the same vorticity that appears in General Relativity? Is vorticity related to the Frame-Dragging of General Relativity?
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The classical fluid mechanical idea of a vortex is well illustrated by this picture, 
which shows a vortex created by the passage of an aircraft wing, revealed by colored smoke.
In classical fluid dynamics, a region of mostly rotational fluid around an axis line is known as a vortex. This axis line can be curved or straight, depending on the physical conditions at play. Vortices form in stirred fluids, such as the picture above, as well as in examples such as boat wake or paddle short lived whirlpools but also including long lived atmospheric phenomena, such as Jupiter's Great Red Spot.
Sources such as this one Gravity B Probe (which I have condensed and slightly altered for brevity) have sometimes used the word vortex in an unintentional but possibly confusing way, such as:
For readers who are not experts in relativity: Geodetic precession is the amount of wobble caused by the static mass of the Earth (the dimple in spacetime) and the frame dragging effect is the amount of wobble caused by the spin of the Earth (the twist in spacetime). Both values are in precise accord with the consequences of Einstein's General Relativity theory and are of value as they add further confirmation for it's correct description of spacetime. The type of spacetime vortex that exists around Earth is duplicated and magnified elsewhere in the cosmos--around massive neutron stars, black holes, and active galactic nuclei.
This General Relativity Rotation and Frame Dragging post is a comprehensive answer to the (totally unrelated classical vortex flow), described above. I lack the expertise to even attempt to summarise it and it is best read after forgetting about classical fluid dynamical vortex phenomena briefly described above.
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Yes frame dragging is related to vorticity; consider Kerr spacetime for simplicity in the following.
In Kerr space-time one has a time-like Killing field $\xi^{\alpha}$. The congruence of observers $u^{\alpha} = \xi^{\alpha}/|\xi|$ following orbits of $\xi^{\alpha}$ are of course the observers who are at rest with respect to the central mass and hence are at rest with respect to the asymptotic Lorentz frame at spatial infinity in which the central mass is at rest. In other words, they have a vanishing angular velocity, $\omega = \frac{d\phi}{dt} = 0$, relative to infinity.
If any one of these observers carries a set of spatial axes $e_i$ that are fixed with respect to infinity, so that the Lie derivative $\mathcal{L}_{\xi}e_i = 0$, then it can be shown that $$F_{u}e^{\alpha}_i = -(\xi_{\delta}\xi^{\delta})^{-1}\epsilon^{\alpha\beta\gamma\sigma}\xi_{\beta}\nabla_{\gamma}\xi_{\sigma} \neq 0$$ where $F_u$ is the Fermi-Walker derivative along $u$; see for example Lightman et al problem 11.10. What this means is even though the spatial axes are fixed with respect to the asymptotic Lorentz frame, they nonetheless precess relative to a set of inertial guidance gyroscopes comoving with the observer. Since $\xi_{[\beta}\nabla_{\gamma}\xi_{\sigma]} \neq 0$ if and only if the space-time is stationary but not static, which intuitively means the space-time is rotating, this is indeed frame dragging
The quantity $$(\xi_{\delta}\xi^{\delta})^{-1}\epsilon^{\alpha\beta\gamma\sigma}\xi_{\beta}\nabla_{\gamma}\xi_{\sigma}$$ is precisely the vorticity (also called twist) of $u^{\alpha}$ i.e. it is the vorticity of the congruence of observers at rest with respect to spatial infinity.
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