Editing Lorentz force (section)

=== 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.