The Lorentz Force

doug <sweetser@world.com>

Introduction
A quaternion equation for the Lorentz force
Implications

Introduction

The Lorentz force acts on a moving charge.  The covariant form of this law is:

(((d Energy)/(d τ)), ((d P)/(d τ))) =  
        γ e (β.E, E  + β × B);

(((d Energy)/(d τ)), ((d P)/(d τ))) =  
        γ e ( β.E, E  + β × B)

In the classical case for a point charge, beta is zero and the E = k e/r^2, so the Lorentz force simplifies to Coulomb's law.  Rewrite this in terms of the potentials phi and A.

\(\(\((\(d\ E\)\/\(d\ τ\), \(d\ P\)\/\(d\ τ\))\)
 = 
      γ\ e 
        \((\(-\ β.\((∂A\/∂t + ∇ϕ)\)\), 
          \(-\((∂A\/∂t + \ ∇ϕ)\)\)\  + \ 
            β\ X \((∇XA)\))\); \)\)

\(\((\(d\ E\)\/\(d\ τ\), \(d\ P\)\/\(d\ τ\))\)
 = \ 
      γ\ e 
        \((\(-\ β.\((∂A\/∂t + ∇ϕ)\)\), 
          \(-\((∂A\/∂t + \ ∇ϕ)\)\)\  + 
            β\ X \((∇XA)\))\)\)

In this notebook, I will look for a quaternion equation that can generate this covariant form of the Lorentz force in the Lorentz gauge.  By using potentials and operators, it may be possible to create other laws like the Lorentz force, in particular, one for gravity.

A quaternion equation for the Lorentz force

The Lorentz force is composed of two parts.  First, there is the E and B fields.  Generate those just as was done for the Maxwell equations

q[(∂/∂)t, 
        (Overscript[▽,  ⇀]) ]. 
      q[ϕ[t], (Overscript[A,  ⇀]) [t]].{1, 0}{-
              (Overscript[▽,  ⇀]) ·(Overscript[A,  
                  ⇀]) [t] + ∂ϕ[t]/∂t, 
        (Overscript[▽,  ⇀])  X 
             (Overscript[A,  ⇀]) [t] + 
          (Overscript[▽,  ⇀]) , 
        ϕ[t] + ∂(Overscript[A,  ⇀]) [t]/∂t}

((∂/∂)t, 
          (Overscript[▽,  ⇀]) ) 
        (ϕ, (Overscript[A,  ⇀]) ) = 
      (-(Overscript[▽,  ⇀]) ·(Overscript[A,  
                  ⇀])  + ∂(ϕ/∂)t, 
          (Overscript[▽,  ⇀])  X 
               (Overscript[A,  ⇀])  + 
            (Overscript[▽,  ⇀]) , 
          ϕ + ∂(((Overscript[A,  ⇀]) )/∂)t)

Another component is the 4-velocity

q[γ,  γ β]

V = (γ,  γ β)

Multiplying these two terms together creates thirteen terms, only 5 of whom belong to the Lorentz force.  That should not be surprising since a bit of algebra was needed to select only the covariant terms that appear in the Maxwell equations.  After some searching, I found the combination of terms required to generate the Lorentz force.

\(\(\(\(\((
                    q[∂\/∂t, 
                      \(-\(▽\& ⇀ \)\)]\)}\).\ 
                q[ϕ[t], \(-\(A\& ⇀ \)[t]\)]]\).q[γ, 
              \(-γβ\)]
	 - \ 
          q[γ, \(-γβ\)].q[∂\/∂t, 
              \(▽\& ⇀ \)].q[ϕ[t], 
              \(A\& ⇀ \)[t]])\).{1, 0}\)\({\(-γβ\)·\((
            \(▽\& ⇀ \), 
              ϕ[t] + ∂\(A\& ⇀ \)[t]\/∂t)
            \), 
	\(-γ\) 
            \((\(▽\& ⇀ \), 
                ϕ[t] + 
                  ∂\(A\& ⇀ \)[t]\/∂t)\) + \ 
          γβ\ X \((
            \(▽\& ⇀ \) X 
              \( A\& ⇀ \)[t])\), }\)

\(\((∂\/∂t, 
            \(-\(▽\& ⇀ \)\))\) 
          \((ϕ, \(-\(A\& ⇀ \)\))\) 
          \((γ, \(-γβ\))\) - \ 
        \((γ, \(-γβ\))\) 
          \((∂\/∂t, 
            \(▽\& ⇀ \))\) 
          \((ϕ, \(A\& ⇀ \))\)
 = 
      \(γ 
          \((\(-β\)·\((
                  \(▽\& ⇀ \), 
                    ϕ + ∂\(A\& ⇀ \)\/∂t)
                  \), \(-\((
                    \(▽\& ⇀ 
                          \), 
                      ϕ + 
                        ∂\(A\& ⇀ \)\/∂t)
                    \)\) + \ 
                β\ X 
                  \((\(▽\& ⇀ \) X 
                      \( A\& ⇀ \))\), )\)
 = \ 
        γ\ e \((\ β.E, E\  + β\ ×\ B)\)\)\)

This combination of differential quaternion operator, quaternion potential and quaternion 4-velocity generates the covariant form of the Lorentz operator in the Lorentz gauge, minus a factor of the charge e which operates as a scalar multiplier.

Implications

By writing the covariant form of the Lorentz force as an operator acting on a potental, it may be possible to create other laws like the Lorentz force.  For point sources in the classical limit, these new laws must have the form of Coulomb's law, F = k e e'/r^2.  An obvious candidate is Newton's law of gravity, F = - G m m'/r^2.  This would require a different type of scalar potential, one that always had the same sign.