Deriving Transformation in Electrodynamics

• Deriving Transformation is a study that formulates electromagnetic field laws under Lorentz transformations using tensorial symmetry arguments.
• It employs a coordinate-free tensor approach to derive boost-induced E–B mixing and to distinguish pure spatial rotations where fields only rotate.
• The findings highlight how symmetry principles simplify derivations, providing clear insights for handling time-varying fields in relativistic electrodynamics.

Deriving electromagnetic field transformations in special relativity denotes the problem of obtaining the electric- and magnetic-field transformation laws under proper Lorentz transformations directly from first principles. A symmetry-based treatment revisits the standard classic special relativistic transformation of the electromagnetic field by deriving the laws for arbitrary oblique uniform relative boost velocities from the antisymmetry of the EM-field tensor and the symmetry of the general Lorentz-boost transformation, and by showing that under proper, instantaneous $3$D rotations the fields transform as a passive rotation of the coordinate axes. The resulting relations are coordinate-free, manifestly encompass time varying fields, and distinguish sharply between boost-induced $E$–$B$ mixing and the absence of such mixing under pure spatial rotation (Bedkihal et al., 2021).

1. Tensorial formulation

The derivation is carried out in units $c=1$ and signature $(+,-,-,-)$. In an inertial frame $\Sigma$, the electromagnetic field is encoded in the antisymmetric tensor

$$F^{\mu\nu} =\begin{pmatrix} 0 & -E^1 & -E^2 & -E^3\ E^1 & 0 & -B^3 & B^2\ E^2 & B^3 & 0 & -B^1\ E^3 &-B^2 & B^1 & 0 \end{pmatrix},$$

with the identifications $F^{0i}=-E^i$ and $F^{ij}=-\varepsilon^{\,ij}{}_{k}\,B^k$. This formulation packages the electric and magnetic fields into a single antisymmetric object and makes the transformation problem a tensorial one rather than a component-by-component exercise based on special source configurations (Bedkihal et al., 2021).

For a boost by the $3$-velocity $E$0, $E$1, the Lorentz matrix is

$E$2

The field tensor then transforms according to

$E$3

Because the boost matrix is symmetric and the field tensor is antisymmetric, the derivation can exploit symmetry directly, preemptively eliminating the labor of performing complicated matrix multiplications (Bedkihal et al., 2021).

2. Arbitrary oblique boosts and the electric field

The electric field in the boosted frame $E$4 is defined by $E$5. Expanding $E$6 using the tensor law yields

$E$7

since the $E$8 term vanishes. Substituting

$E$9

together with the boost components gives an intermediate expression in which the decisive simplification is

$B$0

that is, symmetric$B$1antisymmetric $B$2 (Bedkihal et al., 2021).

After regrouping and using $B$3, one obtains the standard coordinate-free law

$B$4

The essential point is that the derivation is ab initio for arbitrary oblique uniform relative boost velocities. It does not depend on ideal geometries generating special static charge and steady current distributions, nor on introducing parallel and perpendicular field components as a preliminary special-case device. This suggests a cleaner route from the tensor law to the full transformation formula than the traditional generalization from specially chosen configurations (Bedkihal et al., 2021).

3. Magnetic-field transformation and duality

The magnetic-field transformation can be derived in two equivalent ways. The first uses the dual tensor

$B$5

which transforms in the same way as $B$6. Replacing $B$7 and $B$8 in the electric-field law gives

$B$9

The second route derives $c=1$0 directly and contracts with $c=1$1, again using the vanishing of symmetric$c=1$2antisymmetric contractions (Bedkihal et al., 2021).

Taken together, the boost laws are

$c=1$3

These formulas manifestly encompass time varying fields. A common misconception is that the standard field transformations are inherently tied to static source models; the tensor derivation shows instead that the transformation laws follow from Lorentz covariance itself and remain valid without restricting attention to time-independent configurations (Bedkihal et al., 2021).

4. Proper spatial rotations

A proper $c=1$4D rotation about axis $c=1$5 by angle $c=1$6 is represented by the block-diagonal Lorentz matrix

$c=1$7

where $c=1$8. Since $c=1$9, the transformed temporal-spatial components satisfy

$(+,-,-,-)$0

so that

$(+,-,-,-)$1

Under any proper spatial rotation $(+,-,-,-)$2, $(+,-,-,-)$3 and $(+,-,-,-)$4 remain “pure” electric or magnetic fields (Bedkihal et al., 2021).

This result is structurally distinct from the boost case. Under a pure rotation, no $(+,-,-,-)$5–$(+,-,-,-)$6 mixing occurs; the fields transform independently as purely electric and magnetic fields, respectively. The transformation is interpreted as a passive rotation of the coordinate axes. That distinction is conceptually important: boosts probe the spacetime structure linking electric and magnetic sectors, whereas proper spatial rotations act within the spatial representation carried separately by each field (Bedkihal et al., 2021).

5. Kinematic derivations that underpin the transformation laws

The field-tensor derivation presupposes the Lorentz transformation, and the literature represented here shows several distinct ways in which that kinematic structure may itself be derived. One approach begins with the most general linear mapping between two sets of events in two inertial reference frames and uses homogeneity, isotropy, the principle of relativity, the correspondence limit, and the constancy of the speed of light to recover the familiar Lorentz form (Shahandeh, 2013). A second, symmetry-first derivation starts from the Principle of Relativity, supplemented by spacetime homogeneity, isotropy, continuity in the event coordinates, and the absence of a distinguished inertial frame; transformation consistency then yields a one-parameter family characterized by a universal constant $(+,-,-,-)$7, and the light postulate fixes $(+,-,-,-)$8 (Tan, 24 May 2026). A third derivation shows that the structure of the Lorentz transformations follows from the absence of privileged inertial reference frames, group closure, and homogeneity and isotropy, so that an invariant speed need not be presupposed but appears through the constant $(+,-,-,-)$9 in the velocity-addition law $\Sigma$0 (Pelissetto et al., 2015). A fourth route formulates Lorentz transformations as exponentials of rotation and boost generators, introduces rapidity $\Sigma$1 to restore additivity, and recovers the standard boost matrix in terms of $\Sigma$2 and $\Sigma$3 (Durney, 2011).

These derivations clarify the status of the boost matrix used in electromagnetic field transformations. They also delimit its scope. A later result shows that if one imposes only the law of inertia, the most general inertial-frame transformation is nonlinear and projective-linear; imposing isotropic light-speed invariance reduces the transformation to the affine-linear Lorentz case (Agia, 31 Dec 2025). A plausible implication is that the tensor derivation of the electromagnetic field laws is inseparable from the specific affine-linear kinematics singled out by Lorentzian invariance.

6. Scope, interpretation, and methodological significance

The derivation under arbitrary oblique boosts and arbitrary spatial rotations is presented as a didactical exercise amenable to incorporation in graduate courses on relativistic electrodynamics. Its central methodological claim is that powerful symmetry arguments simplify the otherwise tedious calculations while endowing complete generality to the connections obtained. In particular, the antisymmetry of $\Sigma$4 and the symmetry of the general Lorentz-boost transformation remove the need for the standard route through specially engineered charge and current distributions and the subsequent extrapolation to the general case (Bedkihal et al., 2021).

Two interpretive points follow directly. First, the coordinate-free boost laws separate the universal tensorial content of the transformation from any coordinate-specific decomposition into components parallel and perpendicular to the boost. Second, the pure-rotation result corrects the frequent overgeneralization that all proper Lorentz transformations mix electric and magnetic fields: mixing is characteristic of boosts, whereas proper, instantaneous $\Sigma$5D rotations act as passive spatial rotations with

$\Sigma$6

This suggests a concise organizing principle for relativistic electrodynamics: boosts reveal the unified spacetime character of the electromagnetic field tensor, while pure spatial rotations preserve the separate vectorial character of $\Sigma$7 and $\Sigma$8 within a given time slicing (Bedkihal et al., 2021).