---
title: "Covariant Electrodynamics and the Field Tensor"
description: "Collecting the fields into one antisymmetric tensor turns Maxwell's four equations into two and makes the split between electric and magnetic a matter of frame. Charge density and current become a sin"
canonical: https://lightmysky.com/learn/science/covariant-electrodynamics-and-the-field-tensor-mt_RoxM9z6DfK
source: https://lightmysky.com/learn/science/covariant-electrodynamics-and-the-field-tensor-mt_RoxM9z6DfK.md
retrieved: 2026-09-12
---

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# Covariant Electrodynamics and the Field Tensor

Collecting the fields into one antisymmetric tensor turns Maxwell's four equations into two and makes the split between electric and magnetic a matter of frame. Charge density and current become a single four-vector.

Subject: Science · Area: Electricity & Magnetism · Ages 22 to 24
Page: https://lightmysky.com/learn/science/covariant-electrodynamics-and-the-field-tensor-mt_RoxM9z6DfK

## Ready when they can

- Assembles the field tensor from the electric and magnetic components
- Shows how a purely electric field in one frame appears as a mixture in another
- Writes Maxwell's equations in four-vector form and identifies which pair each one covers

## Lesson: One tensor for E and B

All six field components pack into one antisymmetric 4 by 4 tensor F: electric parts sit in the time slots, magnetic parts in the space slots. E and B are not separate entities but two faces of a single object, which is why frames can mix them.

**Example.** Stand beside a resting charge and you measure a pure electric field. Glide past it and the same tensor now shows electric mixed with magnetic. Nothing about the charge changed; your frame did, and the mixture is how tensor components transform.

Maxwell four become two. The antisymmetric-derivative equation covers the source-free pair: no magnetic charge, and changing B makes E curl. The divergence equation ties F to the four-current of charge plus current, covering Gauss and Ampere-Maxwell together.

**Tip.** Charge density and current fuse into one four-vector J. When frames mix, write F and J, transform them, and split back into E and B only at the very end. Splitting early is how sign errors are born.

**Recap.** F unites the fields, frames mix the pieces, Maxwell shrinks to two lines.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Spacetime Intervals and Four-Vectors](https://lightmysky.com/learn/science/spacetime-intervals-and-four-vectors-mt_CYdi7Qpvrf)
- [Displacement Current and Maxwell's Equations](https://lightmysky.com/learn/science/displacement-current-and-maxwells-equations-mt_YR8m8TNfyy)

## Opens up

- [Gauge Freedom and the Potentials as the Working Variables](https://lightmysky.com/learn/science/gauge-freedom-and-the-potentials-as-the-working-variables-mt_BovDzq3WZm)