Transformation of E and B fields under Lorentz boosts

Intuition Beginner

Electric and magnetic fields are not separate entities — they are two aspects of a single electromagnetic field. What looks like a pure electric field to one observer can look like a combination of electric and magnetic fields to a moving observer.

Consider a stationary charge. In its rest frame, it produces a pure electric field (no magnetic field). But if you run past the charge, you see a moving charge — a current — which produces a magnetic field. The magnetic field you observe is created entirely by your motion.

The transformation rules are precise. If you move with velocity $\mathbf{v}$ relative to the source, the fields decompose into components parallel and perpendicular to $\mathbf{v}$. The parallel components are unchanged. The perpendicular components mix: the electric field picks up a contribution from the magnetic field (and vice versa), scaled by a factor of $\gamma =1/\sqrt{1-v^2/c^2}$.

Two quantities are the same for all observers: $E^2-c^2{B}^2$ and $\mathbf{E}\cdot \mathbf{B}$. These are Lorentz invariants. If the electric field dominates in one frame ($E>cB$), it dominates in every frame. You can never transform a purely electric field into a purely magnetic one, or vice versa, unless the field is null ($E=cB$, as for a plane wave).

Visual Beginner

Component	Transformation rule
$E_{\parallel }$	Unchanged
$E_{\perp }$	$\gamma (E_{\perp }+\mathbf{v}\times \mathbf{B})$
$B_{\parallel }$	Unchanged
$B_{\perp }$	$\gamma (B_{\perp }-\mathbf{v}\times \mathbf{E}/c^2)$

Worked example Beginner

A point charge $q$ at the origin has a pure electric field $\mathbf{E}=q\hat{\mathbf{r}}/(4\pi {ϵ}_0{r}^2)$ and $\mathbf{B}=0$.

An observer moves with velocity $\mathbf{v}=v\hat{\mathbf{x}}$ ($v=0.6c$, so $\gamma =1.25$). At the point $(0,1,0)$ (directly above the charge, perpendicular to the motion):

$E_{\parallel }=0$ (E is perpendicular to $\mathbf{v}$ at this point).

$E_{\perp }=\gamma E=1.25\times q/(4\pi {ϵ}_0)$ (the electric field is enhanced by $\gamma$).

$B_{\parallel }=0$. $B_z=-\gamma vE/c^2=-1.25\times 0.6\times E/c=-0.75E/c$.

The moving observer sees a magnetic field. The field pattern looks like an electric field that has been compressed in the direction of motion (by length contraction) plus a magnetic field that circles the direction of motion.

Check your understanding Beginner

Formal definition Intermediate+

Field transformation laws. Under a Lorentz boost with velocity $\mathbf{v}$ (Lorentz factor $\gamma$):

$${\mathbf{E}}_{\parallel }^{\prime }={\mathbf{E}}_{\parallel },{\mathbf{E}}_{\perp }^{\prime }=\gamma ({\mathbf{E}}_{\perp }+\mathbf{v}\times \mathbf{B})$$ $${\mathbf{B}}_{\parallel }^{\prime }={\mathbf{B}}_{\parallel },{\mathbf{B}}_{\perp }^{\prime }=\gamma \left({\mathbf{B}}_{\perp }-\frac{\mathbf{v}\times \mathbf{E}}{c^2}\right)$$

where $\parallel$ and $\perp$ refer to the direction of $\mathbf{v}$.

Covariant derivation. Since $F^{\mu \nu }$ is a rank-2 tensor, it transforms as:

$$F^{\prime \mu \nu }={\Lambda }_{\alpha }^{\mu }{\Lambda }_{\beta }^{\nu }{F}^{\alpha \beta }$$

For a boost along the $x$-axis with velocity $v$, the components of $F^{\mu \nu }$ transform to give the above 3-vector equations. Explicitly, the Faraday tensor:

$$F^{\mu \nu }=\left(\begin{array}{cccc}0& -E_x/c& -E_y/c& -E_z/c\\ E_x/c& 0& -B_z& B_y\\ E_y/c& B_z& 0& -B_x\\ E_z/c& -B_y& B_x& 0\end{array}\right)$$

Under the boost ${\Lambda }_{\nu }^{\mu }$, the transformed components give $E_x^{\prime }=E_x$, $E_y^{\prime }=\gamma (E_y-v{B}_z)$, $E_z^{\prime }=\gamma (E_z+v{B}_y)$, $B_x^{\prime }=B_x$, $B_y^{\prime }=\gamma (B_y+v{E}_z/c^2)$, $B_z^{\prime }=\gamma (B_z-v{E}_y/c^2)$.

Lorentz invariants. Two independent scalars can be formed from $F^{\mu \nu }$:

$$I_1=-\frac{1}{2}{F}_{\mu \nu }{F}^{\mu \nu }=E^2-c^2{B}^2$$ $$I_2=-\frac{1}{4}{ϵ}_{\mu \nu \alpha \beta }{F}^{\mu \nu }{F}^{\alpha \beta }=\mathbf{E}\cdot \mathbf{B}\cdot c$$

Both are the same in every inertial frame. The classification of electromagnetic fields uses these:

• Electric-like ($I_1>0$): $E>cB$ in every frame. Can find a frame where ${\mathbf{B}}^{\prime }=0$.
• Magnetic-like ($I_1<0$): $cB>E$ in every frame. Can find a frame where ${\mathbf{E}}^{\prime }=0$.
• Null ($I_1=0$, $I_2=0$): $E=cB$ and $\mathbf{E}\perp \mathbf{B}$. Plane waves are null fields.

Key derivation Intermediate+

Derivation (Field transformation from the Faraday tensor).

Theorem. Under a boost along the x-axis with velocity v, the electromagnetic fields transform as $E_x^{\prime }=E_x$, $E_y^{\prime }=\gamma (E_y-v{B}_z)$, $B_x^{\prime }=B_x$, $B_y^{\prime }=\gamma (B_y+v{E}_z/c^2)$, and cyclically for the z-components.

Proof. The Lorentz boost along $x$ is:

$${\Lambda }_{\nu }^{\mu }=\left(\begin{array}{cccc}\gamma & -\gamma \beta & 0& 0\\ -\gamma \beta & \gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

where $\beta =v/c$. Compute $F^{\prime \mu \nu }={\Lambda }_{\alpha }^{\mu }{\Lambda }_{\beta }^{\nu }{F}^{\alpha \beta }$:

For $F^{\prime 01}$ (which gives $E_x^{\prime }$):

$$F^{\prime 01}={\Lambda }_{\alpha }^0{\Lambda }_{\beta }^1{F}^{\alpha \beta }={\gamma }^2{F}^{01}-{\gamma }^2{\beta }^2{F}^{10}={\gamma }^2(1-{\beta }^2)F^{01}=F^{01}$$

So $E_x^{\prime }=E_x$.

For $F^{\prime 02}$ (which gives $E_y^{\prime }$):

$$F^{\prime 02}={\Lambda }_0^0{\Lambda }_2^2{F}^{02}+{\Lambda }_1^0{\Lambda }_2^2{F}^{12}=\gamma F^{02}-\gamma \beta F^{12}$$

In terms of fields: $-E_y^{\prime }/c=\gamma (-E_y/c)-\gamma \beta (-B_z)$, giving $E_y^{\prime }=\gamma (E_y-v{B}_z)$.

Similarly for the other components. $\square$

Bridge. The field transformation rules build toward the covariant description of all electromagnetic phenomena in moving frames. The foundational insight is that E and B are not independent fields but components of a single tensor that mix under Lorentz transformations. The central message is that the field invariants $E^2-c^2{B}^2$ and $\mathbf{E}\cdot \mathbf{B}$ classify all electromagnetic fields into electric-like, magnetic-like, and null types. This is exactly the classification used in general relativity to study electromagnetic fields in curved spacetime. Putting these together, the transformation rules generalise the Galilean field transformations (where B is unchanged and E picks up $\mathbf{v}\times \mathbf{B}$), the covariant Larmor formula 10.07.05 uses the boosted fields to compute synchrotron radiation, and the invariants determine the character of the field in any reference frame.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has the Lorentz group and the Faraday tensor but does not contain the explicit E and B transformation rules, the field invariants, or the classification of fields by invariant sign. lean_status: none.

Advanced results Master

The electromagnetic field invariants and classification. The two invariants $I_1=E^2-c^2{B}^2$ and $I_2=\mathbf{E}\cdot \mathbf{B}$ completely classify the electromagnetic field:

• $I_1>0$, $I_2=0$: Electric-like. Exists a frame with pure E-field.
• $I_1<0$, $I_2=0$: Magnetic-like. Exists a frame with pure B-field.
• $I_1=0$, $I_2=0$: Null field. $E=cB$, perpendicular. Plane waves.
• $I_2\ne 0$: Complex field. E and B can never be made parallel or perpendicular by any Lorentz transformation.

The null case ($I_1=I_2=0$) is the most physically important: it includes all electromagnetic radiation fields in the far zone. The fact that a plane wave is null in every frame means that no observer can "cancel" the wave by boosting to a different frame.

The uniformly moving charge. The field of a charge moving with constant velocity is obtained by boosting the Coulomb field. The result is the Heaviside-Feynman formula:

$$\mathbf{E}=\frac{q}{4\pi {ϵ}_0}\frac{(1-{\beta }^2)\hat{\mathbf{R}}}{R^2(1-{\beta }^2{\sin }^2\theta {)}^{3/2}}$$

For $\beta \to 1$ (ultra-relativistic), the field is concentrated in a narrow cone of angular width $\sim 1/\gamma$ around the direction of motion. This "pancake" field is the origin of the beaming effect in synchrotron radiation 10.07.05.

The Thomas precession. When a particle accelerates, the frame that is instantaneously comoving with it precesses relative to the lab frame. This Thomas precession has angular velocity ${\omega }_T=({\gamma }^2/(\gamma +1))(\mathbf{a}\times \mathbf{v})/c^2$. It arises from the non-commutativity of Lorentz boosts and affects the spin-orbit coupling in atomic physics.

Synthesis. The transformation of E and B fields under Lorentz boosts reveals the deep unity of electromagnetism: the electric and magnetic fields are frame-dependent aspects of a single covariant object. The foundational insight is that the field invariants classify all electromagnetic fields into types that are preserved under Lorentz transformations. The central message is that null fields ($E=cB$, $\mathbf{E}\perp \mathbf{B}$) are the hallmark of radiation, and this character is invariant. Putting these together, the field transformations generalise the Galilean limit to relativistic speeds, the boosted Coulomb field produces the "pancake" pattern of ultra-relativistic charges, and the Doppler factor determines the intensity transformation of radiation fields observed from moving sources.

Full proof set Master

Proposition (Invariance of the scalar $\mathbf{E}\cdot \mathbf{B}$). The quantity $\mathbf{E}\cdot \mathbf{B}$ is invariant under all proper Lorentz transformations.

Proof. $\mathbf{E}\cdot \mathbf{B}\cdot c=-\frac{1}{4}{ϵ}_{\mu \nu \alpha \beta }{F}^{\mu \nu }{F}^{\alpha \beta }=-\frac{1}{2}{\tilde{F}}_{\mu \nu }{F}^{\mu \nu }$. Under a proper Lorentz transformation: $F^{\prime \mu \nu }={\Lambda }_{\alpha }^{\mu }{\Lambda }_{\beta }^{\nu }{F}^{\alpha \beta }$ and ${ϵ}_{\mu \nu \alpha \beta }^{\prime }=(\det \Lambda ){\Lambda }_{\mu }^{\rho }{\Lambda }_{\nu }^{\sigma }{\Lambda }_{\alpha }^{\tau }{\Lambda }_{\beta }^{\lambda }{ϵ}_{\rho \sigma \tau \lambda }$. For proper transformations, $\det \Lambda =1$, and the four $\Lambda$ contractions collapse to ${ϵ}_{\rho \sigma \tau \lambda }$, giving $F^{\prime \mu \nu }{F}_{\mu \nu }^{\prime }ϵ$-contracted = $F^{\alpha \beta }{F}_{\alpha \beta }$-contracted. So ${\mathbf{E}}^{\prime }\cdot {\mathbf{B}}^{\prime }=\mathbf{E}\cdot \mathbf{B}$. $\square$

Connections Master

• Faraday tensor 10.06.01 is the covariant object whose components give E and B; the transformation follows from tensor transformation law.
• Lorentz transformations 10.05.01 define the boost under which the fields transform.
• Stress-energy tensor 10.06.03 transforms covariantly along with the field tensor.
• Synchrotron radiation 10.07.05 uses the boosted field pattern of ultra-relativistic charges.
• Larmor formula 10.07.01 is generalised to the covariant form using field transformations.

Historical & philosophical context Master

The transformation of electromagnetic fields was first derived by Lorentz (1904) as part of his theory of the electron. Einstein (1905) derived the same transformation from the postulates of special relativity, showing that the field mixing is a direct consequence of the relativity of simultaneity. Minkowski (1908) showed that E and B are components of a single tensor, providing the most elegant formulation.

The field invariants were identified by Bateman (1910) and Cunningham (1910), who showed that the conformal group (which includes Lorentz transformations) preserves two independent scalars formed from the electromagnetic field. The classification of fields by invariant sign was developed by Synge (1956) and is standard in modern relativistic electrodynamics.

The philosophical significance is that the distinction between "electric" and "magnetic" is frame-dependent, like the distinction between "space" and "time." There is only the electromagnetic field; E and B are its shadow in a particular reference frame.

Bibliography Master

• Lorentz, H. A., "Electromagnetic phenomena in a system moving with any velocity smaller than that of light," Proc. Roy. Acad. Amsterdam 6, 809-831 (1904).
• Minkowski, H., "Die Grundgleichungen fuer die elektromagnetischen Vorgaenge in bewegten Koerpern," Nachr. Ges. Wiss. Goettingen, 53-111 (1908).
• Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
• Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields, 4th ed. (Butterworth-Heinemann, 1975).