the electromagnetic field tensor
Special relativity insists that electric and magnetic fields are not separate things but two faces of one object — what looks purely electric to you can look partly magnetic to someone moving past. The electromagnetic field tensor is the single geometric object that packages E and B together so relativity can act on them cleanly.
It is an antisymmetric rank-2 tensor F^(mu nu), a 4x4 array with F^(mu nu) = -F^(nu mu), built from the four-potential by F^(mu nu) = d^mu A^nu - d^nu A^mu (where d^mu is the four-gradient). Its six independent components are exactly the three components of E and the three of B: in a common convention the time-space entries are the electric field, F^(0i) = E_i / c, and the space-space entries are the magnetic field, F^(ij) = - epsilon_(ijk) B_k. Under a Lorentz boost the whole array transforms as a tensor, which reproduces — automatically and correctly — the mixing rules by which E and B rotate into each other between frames.
Two combinations of F are the same in every frame, the Lorentz invariants F_(mu nu) F^(mu nu) = 2(B^2 - E^2/c^2) and the pseudoscalar built from its dual, proportional to E·B. These state frame-independent facts: if E = cB in one frame (as in a light wave) it holds in all frames; if E and B are perpendicular in one frame they are perpendicular in all. The field tensor is the object that makes Maxwell's equations collapse into two compact, manifestly covariant lines.
A charge at rest makes a purely electric field, F^(0i) = E_i/c with all magnetic components zero. Boost to a frame moving past it and the SAME tensor F, transformed, now has nonzero space-space entries — a magnetic field has appeared. The moving charge is a current, and its magnetism is just the electric field seen from another frame.
One antisymmetric tensor F^(mu nu) holds E (time-space slots) and B (space-space slots); boosting it mixes them.
The split into 'electric' and 'magnetic' is frame-dependent; only the tensor F and its two invariants are absolute. Component signs depend on the metric signature and index conventions, so different textbooks place E and B slightly differently — the physics is in the antisymmetry and the invariants, not in the sign of any one entry.