  The concept of electric and magnetic field lines is intrinsically
non-relativistic. Nonetheless, for certain types of fields satisfying certain
geometric properties, field lines can be defined covariantly. More precisely,
two Lorentz-invariant 2D surfaces in spacetime can be defined such that
magnetic and electric field lines are determined, for any observer, by the
intersection of those surfaces with spacelike hyperplanes. An instance of this
type of field is constituted by the so-called Hopf-Ranada solutions of the
source-free Maxwell equations, which have been studied because of their
interesting topological properties, namely, linkage of their field lines. In
order to describe both geometric and topological properties in a succinct
manner, we employ the tools of Geometric Algebra (aka Clifford Algebra) and use
the Clebsch representation for the vector potential as well as the Euler
representation for both magnetic and electric fields. This description is
easily made covariant, thus allowing us to define electric and magnetic field
lines covariantly in a compact geometric language. The definitions of field
lines can be phrased in terms of 2D surfaces in space. We display those
surfaces in different reference frames, showing how those surfaces change under
Lorentz transformations while keeping their topological properties. As a
byproduct we also obtain relations between optical helicity, optical chirality
and generalizations thereof, and their conservation laws.
