Locally non-rotating observers

[MathJakob]

Hey guys. I have a question about two (possibly ostensibly) different definitions of a locally non-rotating observer that I have come across in my texts.

The first is specifically for stationary, axisymmetric space-times in which we have a canonical global time function [latex]t[/latex] associated with the time-like killing vector field. We define a locally non-rotating observer to be one who follows an orbit of [latex]\nabla^{a}t[/latex] i.e. his 4-velocity is given by [latex]\xi^{a} = (-\nabla^{b}t\nabla_{b}t)^{-1/2}\nabla^{a}t = \alpha \nabla^{a}t[/latex]. Such an observer can be deemed as locally non-rotating because his angular momentum [latex]L = \xi^{a}\psi_{a} = \alpha g_{\mu\nu}g^{\mu}{\gamma}\nabla_{\gamma}t\delta^{\nu}_{\varphi} = \alpha \delta^{\gamma}_{\nu}\delta^{t}_{\gamma}\delta^{\nu}_{\varphi} = \alpha \delta^{t}_{\varphi} = 0[/latex] where [latex]\psi^{a}[/latex] is the axial killing vector field; these observers are also called ZAMOs for this reason. It might also be worth noting that the time-like congruence defined by the family of ZAMOs has vanishing twist [latex]\omega^{a} = \epsilon^{abcd}\xi_{b}\nabla_{c}\xi_{d} = \alpha\epsilon^{ab[cd]}\nabla_{b}t\nabla_{(c}\nabla_{d)}t + \alpha^{3}\epsilon^{a[b|c|d]}\nabla_{(b}t \nabla_{d)}t\nabla^{e}t\nabla_{c}\nabla_{e}t = 0[/latex].


The second definition I have seen is the much more general notion of Fermi-Walker transport. That is, if we choose an initial Lorentz frame [latex]\{\xi^{a}, u^{a},v^{a},w^{a}\}[/latex] and the spatial basis vectors evolve according to [latex]\xi^{b}\nabla_{b}u^{a} = \xi^{a}u_{b}a^{b}[/latex], [latex]\xi^{b}\nabla_{b}v^{a} = \xi^{a}v_{b}a^{b}[/latex], and [latex]\xi^{b}\nabla_{b}w^{a} = \xi^{a}w_{b}a^{b}[/latex], where [latex]a^{b} = \xi^{a}\nabla_{a}\xi^{b}[/latex] is the 4-acceleration, then the observer is said to be locally non-rotating.

My question is, to what extent are these two definitions equivalent (both mathematically and physically) whenever they can both be applied? In other words, in what sense is the qualifier 'rotation' being used in each case?

Let me elucidate my question a little bit. I know that for asymptotically flat axisymmetric space-times, there must exist a fixed rotation axis on which [latex]\psi^{a}[/latex] vanishes. Then the first definition tells us that the ZAMOs have no orbital rotation about this fixed rotation axis (for example no orbital rotation about a Kerr black hole); because these observers are at rest with respect to the [latex]t = \text{const.}[/latex] hypersurfaces, these observers are as close to stationary hovering observers as we can get in a space-time with a rotating source. We know however that such observers have an instrinsic angular velocity [latex]\omega[/latex]; if we imagine a ZAMO holding a small sphere with frictionless prongs sticking out with beads through the prongs then a ZAMO should be able to notice his intrinsic angular velocity [latex]\omega[/latex] by seeing that the beads are thrown outwards along the prongs at any given instant. Is this correct?

Now, on the other hand, the second definition of local non-rotation (using Fermi-Walker transport) seems to be saying sort of the opposite. That is, it seems to be telling us which observers can carry such spheres and never see the beads get thrown outwards i.e. the orientation of the sphere will remain constant (the orientation will be represented by the spatial basis vectors of a Lorentz frame) so these are the observers who have no intrinsic angular velocity i.e. no self-rotation. This is why the two definitions confused me because they seem to be talking about two totally different kinds of non-rotation.