Magnetic Field as the Rotational Component of Electromagnetic Field

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]nmξn —– (1)

θab = h(amhb)nmξn —– (2)

They are smooth fields, orthogonal to ξa in both indices, and satisfy

aξb = ωab + θab + ξammξb) —– (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 ξaaηb. But ξaaηb = ηaaξb. So, by equation (3) and the orthogonality of ξa with ηa at p, we have

ξaaηb = (ωab + θaba —– (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

ξaa(−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 ξbcξd —– (7)

ωab = εabcd ξcωd —– (8)

Hence, ωab = 0 iff ωa = 0.

a = εabcd ξbωcd = εabcd ξb h[crhd]srξs = εabcd ξbhcrhdsrξ

= εabcd ξbgcr gdsrξs = εabcd ξbcξd

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.”


When is the Spacetime Temporally Orientable?


In both general relativity and Newtonian gravitation, forces are represented by vectors at a point. We assume that the total force acting on a particle at a point (computed by taking the vector sum of all of the individual forces acting at that point) must be proportional to the acceleration of the particle at that point, as in F = ma, which holds in both theories. We understand forces to give rise to acceleration, and so we expect the total force at a point to vanish just in case the acceleration vanishes. Since the acceleration of a curve at a point, as determined relative to some derivative operator, must satisfy certain properties, it follows that the vector representing total force must also satisfy certain properties. In particular, in relativity theory, the acceleration of a curve at a point is always orthogonal to the tangent vector of the curve at that point, and thus the total force on a particle at a point must always be orthogonal to the tangent vector of the particle’s worldline at that point.

More precisely, we take a model of relativity theory to be a relativistic spacetime, which is an ordered pair (M, gab), where M is a smooth, connected, paracompact, Hausdorff 4-manifold and gab is a smooth Lorentzian metric. A model of Newtonian gravitation, meanwhile, is a classical spacetime, which is an ordered quadruple (M, tab, hab, ∇), where M is again a smooth, connected, paracompact, Hausdorff 4-manifold, tab and hab are smooth fields with signatures (1, 0, 0, 0) and (0, 1, 1, 1), respectively, which together satisfy tabhbc = 0, and ∇ is a smooth derivative operator satisfying the compatibility conditions ∇atbc = 0 and ∇ahab = 0. The fields tab and hab may be interpreted as a (degenerate) “temporal metric” and a (degenerate) “spatial metric”, respectively. Note that the signature of tab guarantees that locally, we can always find a field ta such that tab = tatb. In the special case where this field can be smoothly extended to a global field with the stated property, we call the spacetime temporally orientable.