Let (M, gab) be the background relativistic spacetime. We are assuming it is temporally orientable and endowed with a particular temporal orientation. Let ξa 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 hab be the spatial projection field determined by ξa. The rotation and expansion fields associated with ξa are defined as follows:
ωab = h[amhb]n ∇mξn —– (1)
θab = h(amhb)n ∇mξn —– (2)
They are smooth fields, orthogonal to ξa in both indices, and satisfy
∇aξb = ωab + θab + ξa(ξm∇mξb) —– (3)
Let γ be an integral curve of ξa, and let p be a point on the image of γ. Further, let ηa be a vector field on the image of γ that is carried along by the flow of ξa and orthogonal to ξa at p. (It need not be orthogonal to ξa elsewhere.) We think of the image of γ as the worldline of a fluid element O, and think of ηa 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 ξa∇aηb. But ξa∇aηb = ηa∇aξb. So, by equation (3) and the orthogonality of ξa with ηa at p, we have
ξa∇aηb = (ωab + θab)ηa —– (4)
at the point. Here we have simply decomposed the relative velocity vector into two components. The first, (ωabηa), is orthogonal to ηa since ωab 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 ωa. It points in the direction of the instantaneous axis of rotation of the fluid. Its magnitude ∥ωa∥ is the instantaneous angular speed of the fluid about that axis. Here ηa connects the fluid element O to the “infinitesimally close” fluid element N. The rotational velocity of N relative to O is given by ωbaηb. The latter is orthogonal to ηa.
In support of this interpretation, consider the instantaneous rate of change of the squared length (−ηbηb) of ηa at p. It follows from equation (4) that
ξa∇a(−ηbηb) = −2θabηaηb —– (5)
Thus the rate of change depends solely on θab. Suppose θab = 0. Then the instantaneous velocity of N with respect to O at p has a vanishing radial component. If ωab ≠ 0, N can still have non-zero velocity there with respect to O. But it can only be a rotational velocity. The two conditions (θab = 0 and ωab ≠ 0) jointly characterize “rigid rotation.”
The rotation tensor ωab 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
ωa = 1/2 εabcd ξbωcd —– (6)
at p. (Here εabcd is a volume element defined on some open set containing p. Clearly, if we switched from the volume element εabcd to its negation, the result would be to replace ωa with −ωa.)
If follows from equation (6) (and the anti-symmetry of εabcd) that ωa is orthogonal to ξa. It further follows that
ωa = 1/2 εabcd ξb∇cξd —– (7)
ωab = εabcd ξcωd —– (8)
Hence, ωab = 0 iff ωa = 0.
2ωa = εabcd ξbωcd = εabcd ξb h[crhd]s ∇rξs = εabcd ξbhcrhds ∇rξs
= εabcd ξbgcr gds ∇rξs = εabcd ξb∇cξd
The second equality follows from the anti-symmetry of εabcd, and the third from the fact that εabcdξb is orthogonal to ξa in all indices.) The equation (6) has exactly the same form as the definition of the magnetic field vector Ba determined relative to a Maxwell field Fab and four-velocity vector ξa (Ba = 1/2 εabcd ξb Fcd). It is for this reason that the magnetic field is sometimes described as the “rotational component of the electromagnetic field.”