Let (M, g_{ab}) be the background relativistic spacetime. We are assuming it is temporally orientable and endowed with a particular temporal orientation. Let ξ^{a} be a smooth, future-directed unit timelike vector field on M (or some open subset of M). We understand it to represent the four-velocity field of a fluid. Further, let h_{ab} be the spatial projection field determined by ξ^{a}. The rotation and expansion fields associated with ξ^{a} are defined as follows:

ω_{ab} = h_{[a}^{m}h_{b]}^{n} ∇_{m}ξ_{n} —– (1)

θ_{ab} = h_{(a}^{m}h_{b)}^{n} ∇_{m}ξ_{n} —– (2)

They are smooth fields, orthogonal to ξ^{a} in both indices, and satisfy

∇_{a}ξ_{b} = ω_{ab} + θ_{ab} + ξ_{a}(ξ^{m}∇_{m}ξ_{b}) —– (3)

Let γ be an integral curve of ξ^{a}, and let p be a point on the image of γ. Further, let η^{a} be a vector field on the image of γ that is carried along by the flow of ξ^{a} and orthogonal to ξ^{a} at p. (It need not be orthogonal to ξ^{a} elsewhere.) We think of the image of γ as the worldline of a fluid element O, and think of η^{a} at p as a “connecting vector” that spans the distance between O and a neighboring fluid element N that is “infinitesimally close.” The instantaneous velocity of N relative to O at p is given by ξ^{a}∇_{a}η^{b}. But ξ^{a}∇_{a}η^{b} = η^{a}∇_{a}ξ^{b}. So, by equation (3) and the orthogonality of ξ^{a} with η^{a} at p, we have

ξ^{a}∇_{a}η^{b} = (ω_{a}^{b} + θ_{a}^{b})η^{a} —– (4)

at the point. Here we have simply decomposed the relative velocity vector into two components. The first, (ω_{a}^{b}η^{a}), is orthogonal to η^{a} since ω_{ab} is anti-symmetric. It is naturally understood as the instantaneous rotational velocity of N with respect to O at p.

*The angular velocity (or twist) vector ω ^{a}. It points in the direction of the instantaneous axis of rotation of the fluid. Its magnitude ∥ω^{a}∥ is the instantaneous angular speed of the fluid about that axis. Here η^{a} connects the fluid element O to the “infinitesimally close” fluid element N. The rotational velocity of N relative to O is given by ω_{b}^{a}η^{b}. The latter is orthogonal to η^{a}. *

In support of this interpretation, consider the instantaneous rate of change of the squared length (−η^{b}η_{b}) of η^{a} at p. It follows from equation (4) that

ξ^{a}∇_{a}(−η^{b}η_{b}) = −2θ_{ab}η^{a}η^{b} —– (5)

Thus the rate of change depends solely on θ_{ab}. Suppose θ_{ab} = 0. Then the instantaneous velocity of N with respect to O at p has a vanishing radial component. If ω_{ab} ≠ 0, N can still have non-zero velocity there with respect to O. But it can only be a rotational velocity. The two conditions (θ_{ab} = 0 and ω_{ab} ≠ 0) jointly characterize “rigid rotation.”

The rotation tensor ω_{ab} at a point p determines both an (instantaneous) axis of rotation there, and an (instantaneous) speed of rotation. As we shall see, both pieces of information are built into the angular velocity (or twist) vector

ω^{a} = 1/2 ε^{abcd} ξ_{b}ω_{cd} —– (6)

at p. (Here ε^{abcd} is a volume element defined on some open set containing p. Clearly, if we switched from the volume element ε^{abcd} to its negation, the result would be to replace ω^{a} with −ω^{a}.)

If follows from equation (6) (and the anti-symmetry of ε^{abcd}) that ω^{a} is orthogonal to ξ^{a}. It further follows that

ω^{a} = 1/2 ε^{abcd} ξ_{b}∇_{c}ξ_{d} —– (7)

ω_{ab} = ε_{abcd} ξ^{c}ω^{d} —– (8)

Hence, ω_{ab} = 0 iff ω^{a} = 0.

2ω^{a} = ε^{abcd} ξ_{b}ω_{cd} = ε^{abcd} ξ_{b} h_{[c}^{r}h_{d]}^{s} ∇_{r}ξ_{s} = ε^{abcd} ξ_{b}h_{c}^{r}h_{d}^{s} ∇_{r}ξ_{s }

= ε^{abcd} ξ_{b}g_{c}^{r} g_{d}^{s} ∇_{r}ξ_{s} = ε^{abcd} ξ_{b}∇_{c}ξ_{d}

The second equality follows from the anti-symmetry of ε^{abcd}, and the third from the fact that ε^{abcd}ξ_{b} is orthogonal to ξ^{a} in all indices.) The equation (6) has exactly the same form as the definition of the magnetic field vector B^{a} determined relative to a Maxwell field F_{ab} and four-velocity vector ξ^{a} (B^{a} = 1/2 ε^{abcd} ξ_{b} F_{cd}). It is for this reason that the magnetic field is sometimes described as the “rotational component of the electromagnetic field.”