I am referring to this paper on the frame-dragging effect in general relativity. In eq.(9) of the paper, the vorticity vector is defined as $$\omega^\alpha=\frac{1}{2}\eta^{\alpha\eta l\lambda} u_\eta u_{l,\lambda}.$$
Then, the non-null components $\omega^r$ and $\omega^\theta$ are calculated in the Kerr spacetime and a quantity $$\Omega=(\omega^\alpha \omega_\alpha)^{1/2}$$ is defined that seems to be an expression for angular velocity that is associated with the Lense-Thirring effect.
However, I couldn't understand the physical interpretation of the vorticity vector and how vorticity is related to the angular velocity of the Lense-Thirring effect. Can someone please help me to clarify this concept and provide relevant references?
In fluid mechanics, vorticity is a measure of the local rotation of a fluid element. It is defined as the curl of the velocity field of the fluid: $$ \boldsymbol{\omega} = \nabla \times \mathbf{v}, $$ where $\mathbf{v}$ is the velocity vector of the fluid and $\boldsymbol{\omega}$ is the vorticity vector.
In general relativity, the concept of vorticity can also be applied to the motion of test particles or spinning test bodies in a curved spacetime. The vorticity of a congruence of timelike or null curves in a spacetime is defined as the antisymmetric part of the covariant derivative of the congruence's tangent vector field: $$ \boldsymbol{\omega}_{\alpha\beta} = \nabla_{[\alpha}u_{\beta]}, $$ where $u_{\alpha}$ is the tangent vector field of the congruence.
The Lense-Thirring effect is a frame-dragging effect in general relativity, which predicts that the rotation of a massive object will "drag" the surrounding spacetime, causing a rotation of inertial frames. This effect can be described by the vorticity of the spacetime, which is related to the angular momentum of the massive object.
In the paper you mentioned, the vorticity vector is used to study the Lense-Thirring effect in the Kerr spacetime, which describes the geometry around a rotating black hole. The non-null components of the vorticity vector are calculated, and a quantity $\Omega = (\omega_{\alpha\beta}\omega^{\alpha\beta})^{1/2}$ is defined, which represents the magnitude of the vorticity. This quantity is related to the angular velocity associated with the Lense-Thirring effect.
In summary, vorticity is a measure of the local rotation of a fluid or spacetime. In the context of general relativity, the vorticity of a spacetime is related to the angular momentum of a massive object and can be used to study the Lense-Thirring effect.