Editing Lorentz force (section)

== Relativistic form of the Lorentz force ==

=== 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
|indent =:
|equation = <math> \frac{\mathrm{d} p^\alpha}{\mathrm{d} \tau} = q F^{\alpha \beta} U_\beta </math>
|cellpadding
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|border colour = #50C878
|background colour = #ECFCF4}}
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>

=== Lorentz force in spacetime algebra (STA) ===

The electric and magnetic fields are [[Classical electromagnetism and special relativity|dependent on the velocity of an observer]], so the relativistic form of the Lorentz force law can best be exhibited starting from a coordinate-independent expression for the electromagnetic and magnetic fields <math>\mathcal{F}</math>, and an arbitrary time-direction, <math>\gamma_0</math>. This can be settled through [[spacetime algebra]] (or the geometric algebra of spacetime), a type of [[Clifford algebra]] defined on a [[pseudo-Euclidean space]],<ref>{{cite web|last=Hestenes|first=David|author-link=David Hestenes|title=SpaceTime Calculus|url=https://davidhestenes.net/geocalc/html/STC.html}}</ref> as
<math display="block">\mathbf{E} = \left(\mathcal{F} \cdot \gamma_0\right) \gamma_0</math>
and
<math display="block">i\mathbf{B} = \left(\mathcal{F} \wedge \gamma_0\right) \gamma_0</math>
<math>\mathcal F</math> is a spacetime [[bivector]] (an oriented plane segment, just like a vector is an [[oriented line segment]]), which has six degrees of freedom corresponding to boosts (rotations in spacetime planes) and rotations (rotations in space-space planes). The [[dot product]] with the vector <math>\gamma_0</math> pulls a vector (in the space algebra) from the translational part, while the wedge-product creates a trivector (in the space algebra) who is dual to a vector which is the usual magnetic field vector. The relativistic velocity is given by the (time-like) changes in a time-position vector {{nowrap|<math>v = \dot x</math>,}} where
<math display="block">v^2 = 1,</math>
(which shows our choice for the metric) and the velocity is
<math display="block">\mathbf{v} = cv \wedge \gamma_0 / (v \cdot \gamma_0).</math>

The proper form of the Lorentz force law ('invariant' is an inadequate term because no transformation has been defined) is simply
{{Equation box 1
|indent =:
|equation = <math> F = q\mathcal{F}\cdot v</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

Note that the order is important because between a bivector and a vector the dot product is anti-symmetric. Upon a spacetime split like one can obtain the velocity, and fields as above yielding the usual expression.

=== Lorentz force in general relativity ===
In the [[general theory of relativity]] the equation of motion for a particle with mass <math>m</math> and charge <math>e</math>, moving in a space with metric tensor <math>g_{ab}</math> and electromagnetic field <math>F_{ab}</math>, is given as

<math display="block">m\frac{du_c}{ds} - m \frac{1}{2} g_{ab,c} u^a u^b = e F_{cb}u^b ,
</math>

where <math>u^a= dx^a/ds</math> (<math>dx^a</math> is taken along the trajectory), <math>g_{ab,c} = \partial g_{ab}/\partial x^c</math>, and <math>ds^2 = g_{ab} dx^a dx^b</math>.

The equation can also be written as

<math display="block">m\frac{du_c}{ds}-m\Gamma_{abc}u^a u^b = eF_{cb}u^b ,</math>

where <math>\Gamma_{abc}</math> is the [[Levi-Civita connection#Christoffel symbols|Christoffel symbol]] (of the torsion-free metric connection in general relativity), or as

<math display="block">m\frac{Du_c}{ds} = e F_{cb}u^b ,</math>

where <math>D</math> is the [[covariant differential]] in general relativity.