The Field Tensor and Covariant Maxwell Equations

The Geometric Unification

Imagine you are describing a rotation in three-dimensional space. You might list the angle and the axis, or you might use a matrix that maps one vector to another. In electromagnetism, we have been treating the electric field \(\vec{E}\) and magnetic field \(\vec{B}\) as separate entities. However, just as a rotation matrix is a single object that contains all the information about how vectors change, the electromagnetic field is a single object that contains all the information about how charges and currents interact with spacetime.

The key insight is that the "time" and "space" parts of the field are not fundamentally different. The electric field is essentially the time-space part of the interaction, while the magnetic field is the space-space part. When we switch to a moving reference frame, the "time" axis tilts relative to the "space" axes. Consequently, what was purely electric in one frame becomes a mixture of electric and magnetic components in another. The tensor formalism captures this mixing automatically. It tells us that \(\vec{E}\) and \(\vec{B}\) are not two separate forces, but two faces of the same coin, determined by the observer's motion through spacetime.

Constructing the Field Tensor

To build the tensor, we first assemble the scalar potential \(\phi\) and vector potential \(\vec{A}\) into a four-vector \(A^\mu\). Following the convention where the time component is first, we define \(A^\mu = (\phi/c, A_x, A_y, A_z)\). The factor of \(c\) ensures all components have the same units.

The electromagnetic field tensor \(F^{\mu\nu}\) is defined as \(F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu\), using the metric \(\eta_{\mu\nu} = \mathrm{diag}(+1,-1,-1,-1)\) and \(x^0 = ct\). This construction guarantees that the tensor is antisymmetric, meaning \(F^{\mu\nu} = -F^{\nu\mu}\). The diagonal elements are zero. The off-diagonal elements contain the field components. Specifically, the time-space block (indices 0 and \(i\)) contains the electric field components, and the space-space block (indices \(i\) and \(j\)) contains the magnetic field components.

This matrix form makes the unification explicit. The electric field appears in the first row and column, while the magnetic field occupies the lower-right \(3 \times 3\) block. The antisymmetry is visually apparent: the matrix is skew-symmetric.

Covariant Maxwell Equations

With the tensor defined, we can write Maxwell's equations in a form that is explicitly invariant under Lorentz transformations. We introduce the four-current \(J^\mu = (c\rho, \vec{J})\), which combines charge density and current density into a single four-vector.

The first of the two covariant equations represents the sourced equations: Gauss's law and the Ampère-Maxwell law. It states that the divergence of the field tensor equals the four-current, scaled by the permeability of free space.

When you expand this single equation for \(\nu = 0\), you recover Gauss's law. When you expand it for \(\nu = 1, 2, 3\), you recover the Ampère-Maxwell law. The second covariant equation represents the homogeneous equations: the absence of magnetic monopoles and Faraday's law of induction. It is a cyclic identity involving three derivatives of the tensor.

This equation is a mathematical identity that holds because \(F_{\mu\nu}\) is derived from a potential \(A_\mu\). It ensures that the magnetic field lines are always closed loops and that a changing magnetic field induces an electric field in a way consistent with relativity.

Automatic Charge Conservation

One of the most profound results of this formulation is that charge conservation is not an independent law; it is a direct consequence of the antisymmetry of the field tensor. In the standard vector formulation, we often state the continuity equation \(\partial \rho / \partial t + \nabla \cdot \vec{J} = 0\) as a separate requirement for consistency.

In the tensor formulation, we take the four-divergence of the sourced equation \(\partial_\mu F^{\mu\nu} = \mu_0 J^\nu\). Because partial derivatives commute, \(\partial_\nu \partial_\mu F^{\mu\nu}\) is the contraction of a symmetric tensor (\(\partial_\nu \partial_\mu\)) with an antisymmetric tensor (\(F^{\mu\nu}\)). The contraction of a symmetric and an antisymmetric tensor is always zero.

Thus, \(\partial_\nu J^\nu = 0\) follows automatically. This demonstrates that the structure of the electromagnetic field enforces the conservation of charge. It is a beautiful example of how geometric constraints in the theory lead to physical conservation laws.