The Electromagnetic Field Tensor

The four-potential carries a gauge redundancy, so it is not itself observable. The observable fields come from a particular antisymmetric combination of its derivatives, one that is invariant under $A^{\mu }\to A^{\mu }+{\partial }^{\mu }\chi$. That combination is a rank-two tensor, and packing the six numbers of $\vec{E}$ and $\vec{B}$ into its components is the step that makes electromagnetism manifestly Lorentz covariant.¹

The conventions of the previous lesson carry over: signature $(-,+,+,+)$, ${\partial }_{\mu }=(\frac{1}{c}{\partial }_t,\nabla )$, and $A^{\mu }=(\varphi /c,\vec{A})$.

Building the field strength

The gauge shift $A^{\mu }\to A^{\mu }+{\partial }^{\mu }\chi$ adds ${\partial }^{\mu }{\partial }^{\nu }\chi$ to ${\partial }^{\mu }{A}^{\nu }$. Because partial derivatives commute, ${\partial }^{\mu }{\partial }^{\nu }\chi ={\partial }^{\nu }{\partial }^{\mu }\chi$, the antisymmetric part is unchanged. Define the electromagnetic field tensor, or field strength, as

Under a gauge transformation,

The field strength is gauge invariant, which is the property that qualifies it to hold physical fields. It is antisymmetric, $F^{\mu \nu }=-F^{\nu \mu }$, so its diagonal vanishes and it has six independent components — exactly the count of $\vec{E}$ and $\vec{B}$ together.

Reading off E and B

Evaluate the components. For the time–space entries, with $A^0=\varphi /c$ and ${\partial }^0=-\frac{1}{c}{\partial }_t$,

The time–space block of $F^{\mu \nu }$ is the electric field, scaled by $1/c$. For the space–space entries, with spatial indices raised trivially,

using $\vec{B}=\nabla \times \vec{A}$. The space–space block is the magnetic field, arranged antisymmetrically. Assembled, the tensor is

The electric field occupies the first row and column; the magnetic field fills the spatial $3\times 3$ block. The single object $F^{\mu \nu }$ holds both fields; electric and magnetic are the names of its time–space and space–space parts in one chosen frame.

Lowering both indices flips the sign of the time–space entries only, since each factor of ${\eta }_{00}=-1$ appears once there and twice (or zero times) elsewhere:

Fields mix under a boost

Because $\vec{E}$ and $\vec{B}$ are components of a single tensor, a change of frame mixes them. Under a Lorentz transformation the field strength transforms as a rank-two tensor,

A boost $\Lambda$ rotates the time axis into a space axis, so it rotates entries of the electric block into the magnetic block and back. A field that is purely electric in one frame acquires a magnetic part in another. The next lesson works out the transformation explicitly; the point here is structural. There is no frame-independent split of the field into electric and magnetic pieces, any more than there is a frame-independent split of a four-vector into time and space. There is only $F^{\mu \nu }$, and observers slice it differently.

The dual tensor

A second antisymmetric tensor is built from $F$ using the totally antisymmetric Levi-Civita symbol ${ϵ}^{\mu \nu \alpha \beta }$ (with ${ϵ}^{0123}=+1$). The dual field tensor is

Contracting with $ϵ$ exchanges the roles of the two blocks. The dual is obtained from $F$ by the substitution

so that its components are

The dual is the tool for writing the two source-free Maxwell equations compactly, as the covariant Maxwell lesson shows: ${\partial }_{\mu }{\tilde{F}}^{\mu \nu }=0$ is Faraday's law and $\nabla \cdot \vec{B}=0$ together. The exchange $\vec{E}\leftrightarrow c\vec{B}$ (with a sign) is the electric–magnetic duality of the source-free equations.

The two invariants

From the six components of the field two Lorentz scalars can be built, and only two independent ones. They are the full contractions of $F$ with itself and with its dual. The first is

The time–space entries contribute $-2E^2/c^2$ (each carries a sign from lowering one time index) and the space–space entries contribute $+2B^2$. The second invariant comes from the dual:

These two combinations,

have the same value in every inertial frame. They classify the field.

• $\vec{E}\cdot \vec{B}$ is invariant. If $\vec{E}$ and $\vec{B}$ are perpendicular in one frame (as in a light wave), they are perpendicular in every frame.
• $E^2-c^2{B}^2$ is invariant. Its sign labels the field as electrically dominated ($E>cB$), magnetically dominated ($E<cB$), or balanced ($E=cB$), the same label in all frames.

Several statements follow at once. If a field is purely electric in some frame, then $\vec{E}\cdot \vec{B}=0$ and $E^2-c^2{B}^2>0$ there, hence in every frame — so no boost can transform it into a purely magnetic field, though it can give it a magnetic part. If both invariants vanish, $\vec{E}\perp \vec{B}$ and $E=cB$ in every frame; this is the invariant signature of a radiation field, a light wave, which no observer can boost away.

Summary

• The field-strength tensor $F^{\mu \nu }={\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu }$ is antisymmetric and gauge invariant. Its six independent components are $\vec{E}$ (time–space block, scaled by $1/c$) and $\vec{B}$ (space–space block).
• A boost transforms $F$ as a tensor and mixes its blocks; the split into electric and magnetic fields is frame dependent, while $F^{\mu \nu }$ is not.
• The dual tensor ${\tilde{F}}^{\mu \nu }=\frac{1}{2}{ϵ}^{\mu \nu \alpha \beta }{F}_{\alpha \beta }$ exchanges $\vec{E}/c\leftrightarrow \vec{B}$ (with a sign) and packages the source-free equations.
• The two invariants $E^2-c^2{B}^2$ and $\vec{E}\cdot \vec{B}$ take the same value in every frame and classify the field; both vanish for a radiation field.