Editing Lorentz force (section)

=== Covariant form of the Lorentz force ===

==== Field tensor ====

{{main|Covariant formulation of classical electromagnetism|Mathematical descriptions of the electromagnetic field}}

Using the [[metric signature]] {{math|(1, −1, −1, −1)}}, the Lorentz force for a charge {{mvar|q}} can be written in [[Lorentz covariance|covariant form]]:{{sfn|Jackson|1998|loc=chpt. 11}}
{{Equation box 1
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|equation = <math> \frac{\mathrm{d} p^\alpha}{\mathrm{d} \tau} = q F^{\alpha \beta} U_\beta </math>
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where {{mvar|p<sup>α</sup>}} is the [[four-momentum]], defined as
<math display="block">p^\alpha = \left(p_0, p_1, p_2, p_3 \right) = \left(\gamma m c, p_x, p_y, p_z \right) ,</math>
{{mvar|τ}} the [[proper time]] of the particle, {{mvar|F<sup>αβ</sup>}} the contravariant [[electromagnetic tensor]]
<math display="block">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}
</math>
and {{mvar|U}} is the covariant [[four-velocity|4-velocity]] of the particle, defined as:
<math display="block">U_\beta = \left(U_0, U_1, U_2, U_3 \right) = \gamma \left(c, -v_x, -v_y, -v_z \right) ,</math>
in which
<math display="block">\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} } }</math>
is the [[Lorentz factor]].

The fields are transformed to a frame moving with constant relative velocity by:
<math display="block"> F'^{\mu \nu} = {\Lambda^{\mu} }_{\alpha} {\Lambda^{\nu} }_{\beta} F^{\alpha \beta} \, ,</math>
where {{math|Λ<sup>''μ''</sup><sub>''α''</sub>}} is the [[Lorentz transformation]] tensor.

==== Translation to vector notation ====

The {{math|1=''α'' = 1}} component ({{mvar|x}}-component) of the force is
<math display="block"> \frac{\mathrm{d} p^1}{\mathrm{d} \tau} = q U_\beta F^{1 \beta} = q\left(U_0 F^{10} + U_1 F^{11} + U_2 F^{12} + U_3 F^{13} \right) .</math>

Substituting the components of the covariant electromagnetic tensor ''F'' yields
<math display="block"> \frac{\mathrm{d} p^1}{\mathrm{d} \tau} = q \left[U_0 \left(\frac{E_x}{c} \right) + U_2 (-B_z) + U_3 (B_y) \right] .</math>

Using the components of covariant [[four-velocity]] yields
<math display="block"> \frac{\mathrm{d} p^1}{\mathrm{d} \tau}
= q \gamma \left[c \left(\frac{E_x}{c} \right) + (-v_y) (-B_z) + (-v_z) (B_y) \right]
= q \gamma \left(E_x + v_y B_z - v_z B_y \right)
= q \gamma \left[ E_x + \left( \mathbf{v} \times \mathbf{B} \right)_x \right] \, .
 </math>

The calculation for {{math|1=''α'' = 2, 3}} (force components in the {{mvar|y}} and {{mvar|z}} directions) yields similar results, so collecting the three equations into one:
<math display="block"> \frac{\mathrm{d} \mathbf{p} }{\mathrm{d} \tau} = q \gamma\left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right) , </math>
and since differentials in coordinate time {{mvar|dt}} and proper time {{mvar|dτ}} are related by the Lorentz factor,
<math display="block">dt=\gamma(v) \, d\tau,</math>
so we arrive at
<math display="block"> \frac{\mathrm{d} \mathbf{p} }{\mathrm{d} t} = q \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right) .</math>

This is precisely the Lorentz force law, however, it is important to note that {{math|'''p'''}} is the relativistic expression,
<math display="block">\mathbf{p} = \gamma(v) m_0 \mathbf{v} \,.</math>