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.”

Thermodynamics of Creation. Note Quote.


Just like the early-time cosmic acceleration associated with inflation, a negative pressure can be seen as a possible driving mechanism for the late-time accelerated expansion of the Universe as well. One of the earliest alternatives that could provide a mechanism producing such accelerating phase of the Universe is through a negative pressure produced by viscous or particle production effects. The viscous pressure contributions can be seen as small nonequilibrium contributions for the energy-momentum tensor for nonideal fluids.

Let us posit the thermodynamics of matter creation for a single fluid. To describe the thermodynamic states of a relativistic simple fluid we use the following macroscopic variables: the energy-momentum tensor Tαβ ; the particle flux vector Nα; and the entropy flux vector sα. The energy-momentum tensor satisfies the conservation law, Tαβ = 0, and here we consider situations in which it has the perfect-fluid form

Tαβ = (ρ+P)uαuβ − P gαβ

In the above equation ρ is the energy density, P is the isotropic dynamical pressure, gαβ is the metric tensor and uα is the fluid four-velocity (with normalization uαuα = 1).

The dynamical pressure P is decomposed as

P = p + Π

where p is the equilibrium (thermostatic) pressure and Π is a term present in scalar dissipative processes. Usually, it is associated with the so-called bulk pressure. In the cosmological context, besides this meaning, Π can also be relevant when particle number is not conserved. In this case, Π ≡ pc is called the “creation pressure”. The bulk pressure,  can be seen as a correction to the thermostatic pressure when near to equilibrium, thus, it should be always smaller than the thermostatic pressure, |Π| < p. This restriction, however, does not apply for the creation pressure. So, when we have matter creation, the total pressure P may become negative and, in principle, drive an accelerated expansion.

The particle flux vector is assumed to have the following form

Nα = nuα

where n is the particle number density. Nα satisfies the balance equation Nα = nΓ, where Γ is the particle production rate. If Γ > 0, we have particle creation, particle destruction occurs when Γ < 0 and if Γ = 0 particle number is conserved.

The entropy flux vector is given by

sα = nσuα

where σ is the specific (per particle) entropy. Note that the entropy must satisfy the second law of thermodynamics sα ≥ 0. Here we consider adiabatic matter creation, that is, we analyze situations in which σ is constant. With this condition, by using the Gibbs relation, it follows that the creation pressure is related to Γ by

pc = − (ρ+p)/3H Γ

where H = a ̇/a is the Hubble parameter, a is the scale factor of the Friedmann-Robertson-Walker (FRW) metric and the overdot means differentiation with respect to the cosmic time. If σ is constant, the second law of thermodynamics implies that Γ ≥ 0 and, as a consequence, particle destruction (Γ < 0) is thermodynamically forbidden. Since Γ ≥ 0, it follows that, in an expanding universe (H > 0), the creation pressure pc cannot be positive.