Magnetic Field as the Rotational Component of Electromagnetic Field

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^mh_b]ⁿ ∇_mξ_n —– (1)

θ_ab = h_(a^mh_b)ⁿ ∇_mξ_n —– (2)

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

∇_aξ_b = ω_ab + θ_ab + ξ_a(ξ^m∇_mξ_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∇_aη^b. But ξ^a∇_aη^b = η^a∇_aξ^b. So, by equation (3) and the orthogonality of ξ^a with η^a at p, we have

ξ^a∇_aη^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^bη^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^aη^b. The latter is orthogonal to η^a. 

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

ξ^a∇_a(−η^bη_b) = −2θ_abη^aη^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_bω_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∇_cξ_d —– (7)

ω_ab = ε_abcd ξ^cω^d —– (8)

Hence, ω_ab = 0 iff ω^a = 0.

2ω^a = ε^abcd ξ_bω_cd = ε^abcd ξ_b h_[c^rh_d]^s ∇_rξ_s = ε^abcd ξ_bh_c^rh_d^s ∇_rξ_s 

= ε^abcd ξ_bg_c^r g_d^s ∇_rξ_s = ε^abcd ξ_b∇_cξ_d

The second equality follows from the anti-symmetry of ε^abcd, and the third from the fact that ε^abcdξ_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.”

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