structural:reformatting, stylistic:tone, structural:scholar_edition
Field tensor

Using the metric signature (1, −1, −1, −1), the Lorentz force for a charge q can be written in covariant form:

d p α d τ = q F α β U β {\displaystyle {\frac {\mathrm {d} p^{\alpha }}{\mathrm {d} \tau }}=qF^{\alpha \beta }U_{\beta }}

where p is the four-momentum, defined as p α = (p 0, p 1, p 2, p 3) = (γ m c, p x, p y, p z), {\displaystyle p^{\alpha }=\left(p_{0},p_{1},p_{2},p_{3}\right)=\left(\gamma mc,p_{x},p_{y},p_{z}\right),} τ the proper time of the particle, F the contravariant electromagnetic tensor F α β = (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) {\displaystyle F^{\alpha \beta }={\begin{pmatrix}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{pmatrix}}} and U is the covariant 4-velocity of the particle, defined as: U β = (U 0, U 1, U 2, U 3) = γ (c, − v x, − v y, − v z), {\displaystyle U_{\beta }=\left(U_{0},U_{1},U_{2},U_{3}\right)=\gamma \left(c,-v_{x},-v_{y},-v_{z}\right),} in which γ (v) = 1 1 − v 2 c 2 = 1 1 − v x 2 + v y 2 + v z 2 c 2 {\displaystyle \gamma (v)={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-{\frac {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}{c^{2}}}}}}} is the Lorentz factor.

The fields are transformed to a frame moving with constant relative velocity by: F ′ μ ν = Λ μ α Λ ν β F α β, {\displaystyle F'^{\mu \nu }={\Lambda ^{\mu }}_{\alpha }{\Lambda ^{\nu }}_{\beta }F^{\alpha \beta }\,,} where Λ is the Lorentz transformation tensor.