More generally we postulate a 1-vector a_p = A_p + c^-1V_pe₄ satisfying Ñ_pÙa_p = -f_p for irrotational pure bivector 2-field f_p. so that
    m_p = m^-1 Ñ_pf_p = -m^-1 Ñ_p(Ñ_pÙa_p) = -m^-1 Ñ_p¿(Ñ_pÙa_p) = -m^-1 (Ñ_p²a_p - Ñ_p(Ñ_p¿a_p) ) .
    Insisting further that Ñ_p¿a_p = 0  (the Lorentz condition) provides Ñ_p²a_p = -mm_p .

    A 1-potential is said to be e₄-static or static if Ð_e4a_p » 0 " p so that Ð_e4 derivatives vanish. Physicists also sometimes refer to potentials of the form a_p = a⁴e₄ as static. Such an a_p is seen as entirely nonmagnetic (ie. electrostatic) by e₄-observers. Physicists refer to 1-potentials of the form a_p = a(|^_e4(p-c)|) as central so that a static central potential has the form a_p = a⁴(|^_e4(p-c)|)e₄ , a 4D 1-potential formed by using a 3D 0-potential as the e₄ coordinate of an otherwise zero 1-vector. A notable example is the Lienard-Wiechart 1-potential a_p = |e⁴Ùp|^-1 e₄ = |P|^-1 e₄     (where p=P + p⁴e₄) which has Ñ_pa_p = -|P|^-3 Pe₄ = -|P|^-4 (PÙe₄)^~     and     Ñ_p²a_p = 0 (since N=4).

    If the e₄-magnetic (within e₁₂₃) component b_p of a Faraday 2-field f_p = e_p - b_p is non-solenoidal ( Ñ_p¿b_p ¹ 0) at p we say that p is a magnetic source. Other than at such a source, Ñ_p¿b_p = 0 and we can construct a 3D 1-potential 1-vector A_p with Ñ_p[e123]A_p = Ñ_p[e123]ÙA_p = b_p .
    We then have e_p = f_p + b_p     =   -Ñ_pa_p + Ñ_p[e123]A_p   =   -(Ñ_p[e123]a⁴)e₄ - Ñ_p[e4]A_p   =   -(Ñ_p[e123]a⁴)e₄ - e⁴(Ð_e4A_p)
[  Rindler[7.49] has Ñ_p²a_p=4pc^-1 j_p in Â_1,3, Lounesto[13.1] has Ñ_pa_p = -f_p in Â_3,1 ]
    A flow satisfying Ñ_p²m_p = -lm_p "meets" its own 1-potential.

    If f_p is a scalar field satisfying Ñ_p²f_p = lf_p then 1-vector f_pu_p satisfies Ñ_p²f_pu_p = lf_pu_p for any u_p satisfying Ñ_pu_p=0 such as u_p = u or u_p = p|p|^-N .
    If f_p satisfies Ñ_p² f_p = lf_p then f_pd satisfies Ñ_p²f_pd = lf_pd for any constant 1-vector d.
    If f_p satisfies Ñ_p² f_p = lf_p (such as f(x)=f(|x|) where f"(r) +(N-1)r^-1f'(r) = ±lf(r) according as x² is ±) then f_pd solves the Laplace Equation for any p-independant d.
    If f(r) satisfies f"(r) - (N-1)r^-3f(r) + (N-1)r^-1f'(r) = lf(r) then f(|x|)x^~ satisfies Ñ_x² f(|x|)x^~ = l f(|x|)x^~ .
    Thus our condition for a static potential is f"(r) + (N-1)r^-1f'(r) -/+ lf(r) = 0 which we have already seen solved with f(r)=l sin(ar^g)r_b while for radial potential f(r)x^~ we need
    f"(r) + (N-1)r^-1f'(r) + (l - (N-1)r^-3)f(r) = 0 .

Bivector Force Fields
    If p'(t)² is constant then W_p = -p"(t)p'(t) = -p"(t)Ùp'(t) is a pure plussquare bivector . In electrodynamics physicists infer from a trajectory curve solving p" = W_p.pdt the existance of a pervading Faraday electromagnetic  bivector force field quantifiable (along p(t)) as
    f_p(t) = (m₀/q₀)W(t) + b₂(p(t)) where arbitary spacelike 2-blade b₂ Î pdt^* cannot be quantified (deduced) from the trajectory ; and proper mass m₀ and proper charge q₀ are two scalars "associated" with the particle.
    The minus sign in W_p = -p"(t)p'(t) is necessary to accomodate the negative signature of pdt(t) . It does not arise in Â_1,3 timespace.

    If we instead use p" = (W_ppdt)_<1> we can set W_p = W_p plus an arvbitary <0;4>-vector.

    In the case of the helical trajectory p(t) = (½w(1-R²w²)^-½t e₂₁)^↑_§(e₃) we have p"(t)pdt(t)   =   -(1-R²w²)^-3/2w²R R_t§(Rwe₂₁ + e₄₁)     so (p"(t)pdt(t))²   =   (1-R²w²)^-2w⁴R² .

    Bivector field f_p decomposeses into electrostatic and magnetic components = c^-1e_p + b_p where b_p = Be₁₂₃^-1 lies in Îe₁₂₃ and e=Ee₄ has e₄Ùe_p=0

    We "incorporate the local medium" by defining electric displacement Dp = epEp and magnetic intensity spacial bivector hp = mp-1bp where linear dilectrics ep and mp-1 are unitless geometric parameters representing the medium. Scalars in the case of a isotropic medium ; p-independant in a (spaciotemporally) homogeneous medium in which case they are known as the permitivity and pemeability constants respectively.   Units can be chosen to make ep = mp = 1 in a "vacuum". In S.I. units we have
    e0 = (2hca)-1Qe-2 = (hc)-1(2p) » 8.8541878176×10-12 F m-1 = (4p)-1 where one farad = 1 F=1 C V-1 = 1 m-2 kg-1 s4 A2 = 1 m-2 kg-1 s2 C2 = 10-7c2 m ; and m0 = 4p×10-7 H m-1 = 4p where one henri = 1 H = 1 m2 kg s-2 A-2 = 1 m2 kg C-2 = 107 m .

    Bivector field g_p = cD_pe₄ - H_pe₁₂₃ = ce_pe_p - m_p^-1b_p provides a geometric analog of the Minkowski tensor form of the Maxwell equations
    Ñ_p¿g_p = j_p    ;     Ñ_pÙf_p = 0
which, if g_p = m^-1f_p for some p-independant geometric linear "dilectric" multiplier m^-1,  are the 1 and 3 grade components of the single geometric Maxwell equation
    Ñ_pf_p   =   Ñ_p¿f_p   =   m j_p .

    Splitting the "dilectric" into e₄-relative scaling components feels nonrelativistic, and it is more natural to replace g_p = ce_pe_p - m_p^-1b_p with g_p = m_p(f_p) where m_p() is a 2-tensor transformation mapping bivectors to bivectors, point-dependant except in a homogenous medium. [ Maxwell's original equations were Ñ×H = J + ¶D/¶t    ; Ñ×E = - ¶B/¶t    ; Ñ.D = r    ; Ñ.B = 0     where H,E,D,and B are spacial 3D 1-vectors in e₄^*   ]
    It is thus natural to regard bivector force field f_p as more fundamental than the flow j_p = m_p^-1 Ñ_p¿f_p , which is fully defined by f_p and    satifisfies "conservation law" Ñ_p¿j_p as an inevitable consequence of its construction.
    If the 1-potential a_p satisies Ñ_p¿a_p=0 then we have Ñ_pa_p=-f_p and hence
    j_p   =   m_p^-1 Ñ_p(f_p)   =   -m_p^-1 Ñ_p² a_p   =   -m_p^-1 Ñ_p¿(Ñ_pÙa_p)     with m_p^-1=m^-1 a point-independant scalar in an isotrophic linear medium. . [  In Â_1,3 we have Ñ_p²a_p = mj_p ]

    In a classical vacuum j_p=0 and so Ñ_pf_p = 0 which has pure Â_3,1 bivector circularly polarised wave solution
    f_p = ak((k_*p)e₁₂₃₄)^↑     for any null 1-vector k and 1-vector a normal to k where a^↑ º e^a denotes exponentiation. 4-blade e₁₂₃₄ is available as a commuting i here because f_p is even.
    Picking a frame with k=w(e₃+e₄) we observe that ke₃₄ = k so that k((k_*p)e₁₂₃₄)^↑ = k((k_*p)e₁₂)^↑ and we can express our solution as
    f_p = a(e₃+e₄)((w(e₃+e₄)_*p)e₁₂)^↑   =   re₁(e₃+e₄)((q+(w(e₃+e₄)_*p))e₁₂)^↑   =   re₁(e₃+e₄)((q+(w(e₃+e₄)_*p))e₁₂₃₄)^↑     where a = re₁e^qe₁₂ is an arbitary 1-vector .
    We say f_p has scalar magnitude r>0, frequency w, and phase q. w>0 is known as right-circular polarisation whereas w<0 gives left-circular polarisation.

    More generally f_p = ((k_*p)^ab)^↑ ka has Ñ_pf_p=0 for any null 1-vector k and constant a, b if a¹0 and k commutes with b.
    Thus if F_p is any solution to Ñ_pF_p = lF_p that commutes with null 1-vector k then F_pk((p¿k)^-2j)^↑ for integer j³1 is also solves Ñ_pF_p = lF_p and by taking j sufficiently large we can ensure F_p is normalisable over nullcnes. by right multiplication by k((k_*p)^-2jb)^↑ for integer j³1, null 1-vector k, and any b commuting with k, eg. b=1 to obtain another solution .

    Such a bivector field f_p is perceived by a (4-frame) observer E as
    f(t) = ^_e4(t)^*(f(p(t))) + ¯_e4(t)^*(f(p(t))) = ^_e4(t)^*(f(p(t))) - (¯_e4(t)^*(f(p(t)))^*)^* = E_E(p(t))e₄ - B_E(p(t))e₁₂₃
where e₁₂₃ 1-vectors E_E(p(t)) º ^_e4(t)^*(f(p(t)))e₄^-1 and B_E(p(t)) º - ¯_e4(t)^*(f(p(t))) e₁₂₃^-1 are known as the relative electric field and the relative magnetic field respectively.
    Relative magnetic fields are more naturally represented by e₄^* bivector b_p º b_E(p(t)) = B_pe₁₂₃ giving f_p = c^-1e_p - b_p but such decompositions are frequently more confusing than useful and it is better to consider the unified field f_p.

    In particular, ¯_e4^*(p") = (q₀/m₀) ¯(f,e₄^*) = (q₀/m₀)e_A so a charge at rest is accelerated only by its relative electric field .

Path tsphere Intersections
    Suppose we have a particle on a timelike worldline Q={q(t) : tÎ[t₁,t₁]} . For a given point p, where does the path Q intercept geometrically noninvertible L^-_p or invertible O^-_p,h0. It can cross L^-_p or O^-_p,h0 at most once if q'(t) is always strictly timelike , and if we assume the particle to have existed for long enough (t₁ suffiently "early") it must cross somewhere so O^-_p,h0 Ç Q is a 1-horoblade correseponding to a   single embedded event q(t_p).
    Let t be the natural parameterisation so that v(t) = v_p º q'(t) is unit timelike " tÎ[t₁,t₁]. We thus have a scalar field t_p=t(p) returning the particle-clock time of the unique event of particle traversal of the rear nullcone L^-_p.
    Define null or backward timelike 1-field  s(p) º s_p º  q(t(p))-p with s_p²=-h₀² and unit timelike 1-field v_p º v(t_p) .

    Euclidean preconceptions might lead us to expect Ñ_p t(p) to be parallel to v_p=v(t) but instead we have  
    Ñ_p t_p = s_p (s_p¿v_p)^-1     giving s_p¿(Ñ_pt_p) = 0    ;     and v_p¿(Ñ_pt_p) = 1 .
[ Proof : Ñ_p(s_pÑ¿s_p)   =   Ñ_p((q(t(p))-p)_Ñ¿s_p)     =   Ñ_p(q(t(p)_Ñ¿s_p) - Ñ_p(p_Ñ¿s_p)   =   (Ñ_pt(p))v_p¿s_p - s_p
    But Ñ_p(s_p²)=Ñ_p(h₀²)=0 so (Ñ_pt(p))v_p¿s_p = s_p. _{[ GAfp 7.84 ]}  .]

    The undirected scalar-funneled chain rule gives
    Ñ_p s_p = Ñ_p(q(t_p)-p) = (Ñ_pt_p)v_p - N = (s_p¿v_p)^-1 s_pv_p - N (s_p¿v_p)^-1 s_pv_p - N .
    Ñ_p v_p = (Ñ_pt_p)v_p' = (s_p¿v_p)^-1 s_pv_p' .

       Ñ_p (s_p¿v_p)   =   -(s_p¿v_p)^-1 ( s_p(s_pÙv_p') + v_p(s_pÙv_p)) = -½(s_p¿v_p)^-1 ( v_ps_pv_p + s_p²v_p' - v_p²s_p - s_pv_p's_p)
[ Proof : From a¿(bÙc)=(a¿b)c-(a¿c)b we have
    Ñ_p(s_p¿v_p)    = (s_p¿Ñ)v_pÑ + (v_p¿Ñ)s_pÑ - s_p¿(Ñ_pÙv_p) - v_p¿(Ñ_pÙs_p) = Ð_spv_p + Ð_vps_p - s_p¿(Ñ_pÙv_p) - v_p¿(Ñ_pÙs_p) .
    The directed scalar-funnelled rule   Ð_aF(g(X)) = F'(g(X))g^Ñ(a)=(a_*(Ñ_xg(X))) F'(g(x)) enables verification that Ð_vps_p = Ð_spv_p = 0   so we have
    Ñ_p(s_p¿v_p)    = - s_p¿(Ñ_pÙv_p) - v_p¿(Ñ_pÙs_p) = - s_p¿(Ñ_pv_p)_<2> - v_p¿(Ñ_ps_p)_<2>
    = -(s_p¿v_p)^-1 ( s_p¿(s_pÙv_p') + v_p¿(s_pÙv_p)) = -(s_p¿v_p)^-1 ( s_p(s_pÙv_p') + v_p(s_pÙv_p))
    = -(s_p¿v_p)^-1 ½( s_p(s_pv_p'-v_p's_p) + v_p(s_pv_p-v_ps_p))  .]

    Nonnegative (since v_p is forward and s_p backward) scalar s_p¿v_p is physically important as the spacial distance (as percieved by the particle at q_p) from q_p to p. It vanishes for null s_p only when v_p is also null and parallel to s_p, , whereupon s_p¿(v_pÙv_p') vanishes. If s_p=-_gu(e₄+_Udp) is backward timelike,  sp¿v_p is minimised by timelike v_p =g_V(e₄+V_p) as _gug_V(1-V_p²_Udp²) where _Udp is opposite parallel to V_p.

Propagated Lienard-Wiechart Potential
    Suppose a timelike (or null) trajectory particle at q_p with unit (or null) four-velocity v_p and acceleration v_p' instantaneously generates a timelike (or null) 1-potential a_p = f(s_p¿v_p) v_p where f: ®Â. We have
    f_p   =   -Ñ_p a_p   =   -(s_p¿v_p)^-1 ( f'(s_p¿v_p)( -s_p(s_pÙv_p')v_p + v_p²(s_pÙv_p)) + f(s_p¿v_p)s_pv_p' )
[ Proof : Ñ_p(f(s_p¿v_p)v_p)   =   (Ñ_pf(s_p¿v_p))v_p +  f(s_p¿v_p) (Ñ_pv_p)   =   f'(s_p¿v_p)(Ñ_p(s_p¿v_p))v_p + f(s_p¿v_p)(s_p¿v_p)^-1 s_pv_p'
= -f'(s_p¿v_p)(s_p¿v_p)^-1 (s_p(s_pÙv_p')+v_p(s_pÙv_p))v_p + f(s_p¿v_p)(s_p¿v_p)^-1 s_pv_p' =   (s_p¿v_p)^-1( f'(s_p¿v_p)( -s_p(s_pÙv_p')v_p+v_p²(s_pÙv_p)) + f(s_p¿v_p)s_pv_p') .]

    Setting f(s_p¿v_p) = a(h₀-(s_p¿v_p)²)^-½ for some p-independant fixed scalar "mass" or "charge" or "ammount" a we have
    f_p = -Ñ_pa_p =   -(s_p¿v_p)^-1 ( -½((h₀-(s_p¿v_p)²)^-3/2  (s_p¿v_p)( -s_p(s_pÙv_p')v_p + v_p²(s_pÙv_p)) + (h₀-(s_p¿v_p)²)^-½ s_pv_p' )
    =   -(s_p¿v_p)^-1 (h₀-(s_p¿v_p)²)^-3/2( -½(  (s_p¿v_p)( -s_p(s_pÙv_p')v_p + v_p²(s_pÙv_p)) + (h₀-(s_p¿v_p)²)s_pv_p' )

    For f(s_p¿v_p) = a(s_p¿v_p)^-1 corresonding to a Lienard-Wiechart potential a |^_vp(s_p)|^-1 v_p for null s_p we have
    a_p = a (s_p¿v_p)^-1 v_p = -a (¯_vp(s_p))^-1 and
    f_p   =   -Ñ_p a_p   =   a (s_p¿v_p)^-3( v_p²(s_pÙv_p) + (s_pv_p')×(s_pv_p) )
[ Proof : Ñ_p((s_p¿v_p)^-1)v_p)   =   (Ñ_p((s_p¿v_p)^-1))v_p +  (s_p¿v_p)^-1 (Ñ_pv_p)   =   -(s_p¿v_p)^-2(Ñ_p(s_p¿v_p))v_p +  (s_p¿v_p)^-2 s_pv_p'
    = (s_p¿v_p)^-3(½( v_ps_pv_p + s_p²v_p' - v_p²s_p -s_pv_p's_p)v_p + (s_p¿v_p) s_pv_p')
    = ½(s_p¿v_p)^-3( v_p²v_ps_p + s_p²v_p'v_p - v_p²s_pv_p - s_pv_p's_pv_p + (s_pv_p+v_ps_p)s_pv_p')
    = (s_p¿v_p)^-3( s_p²(v_p¿v_p') + v_p²(v_pÙs_p) + (s_pv_p)×(s_pv_p') )   =   (s_p¿v_p)^-3( v_p²(v_pÙs_p) + (s_pv_p)×(s_pv_p') )  .]
    Both terms of Ñ_pa_p are pure bivectors so Ñ_p¿a_p = 0, as required of a 1-potential, for any trajectory q(t). Note that
      (s_pv_p')×(s_pv_p)   =   (s_pv_p')(s_pv_p) - (s_pv_p')¿(s_pv_p)     =   (s_pv_p's_p)v_p - (s_pv_p's_p)¿v_p     =   (s_pv_p's_p)Ùv_p   =   -(s_pv_ps_p)Ùv_p'

    When n²=-1 we have -nxn = ¯_n(x) - ^_n(x) so (s_pv_p')×(s_pv_p) =   -((¯_sp(v_p')-^_sp(v_p'))Ùv_p =   ((¯_sp(v_p)-^_sp(v_p))Ùv_p' .   =   (s_pv_p')(s_pv_p) - (s_pv_p's_p</