More generally we postulate a 1-vector
a_{p}
= A_{p} + c^{-1}V_{p}e_{4}
satisfying Ñ_{p}Ùa_{p} = -f_{p} for irrotational pure bivector 2-field f_{p}.
so that
m_{p} = m^{-1} Ñ_{p}f_{p} =
-m^{-1} Ñ_{p}(Ñ_{p}Ùa_{p}) =
-m^{-1} Ñ_{p}¿(Ñ_{p}Ùa_{p}) =
-m^{-1} (Ñ_{p}^{2}a_{p} - Ñ_{p}(Ñ_{p}¿a_{p}) ) .
The tighter Lorentz condition Ñ_{p}¿a_{p} = 0 provides Ñ_{p}^{2}a_{p} = -mm_{p} .
A 1-potential is said to be e_{4}-static or static if Ð_{e4}a_{p} » 0 " p
so that Ð_{e4} derivatives vanish.
Physicists also sometimes refer to potentials of the form a_{p} = a^{4}_{p}e_{4}
as static. Such an a_{p} is seen as entirely nonmagnetic (ie. electrostatic) by e_{4}-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} = a^{4}(|^_{e4}(p-c)|)e_{4} , a 4D 1-potential formed by
using a 3D 0-potential as the e_{4} coordinate of an otherwise zero 1-vector.
A notable example is the Lienard-Wiechart 1-potential
a_{p} = |e^{4}Ùp|^{-1} e_{4} = |P|^{-1} e_{4}
(where p=P + p^{4}e_{4})
which has
Ñ_{p}a_{p} = -|P|^{-3} Pe_{4}
= -|P|^{-4} (PÙe_{4})^{~} and
Ñ_{p}^{2}a_{p} = 0 (since N=4).
If the e_{4}-magnetic (within e_{123}) 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_{p}a_{p} + Ñ_{p}_{[e123]}A_{p}
= -(Ñ_{p}_{[e123]}a^{4})e_{4} - Ñ_{p}_{[e4]}A_{p}
= -(Ñ_{p}_{[e123]}a^{4})e_{4} - e^{4}(Ð_{e4}A_{p})
[ Rindler[7.49] has Ñ_{p}^{2}a_{p}=4pc^{-1} j_{p} in Â_{1,3},
Lounesto[13.1] has Ñ_{p}a_{p} = -f_{p} in Â_{3,1}
]
A flow satisfying Ñ_{p}^{2}m_{p} = -lm_{p} "meets" its own 1-potential.
If f_{p} is a scalar field satisfying Ñ_{p}^{2}f_{p} = lf_{p} then 1-vector
f_{p}u_{p} satisfies Ñ_{p}^{2}f_{p}u_{p} = lf_{p}u_{p} for any monogenic 1-vector u_{p} satisfying
Ñ_{p}u_{p}=0 such as u_{p} = u or u_{p} = p|p|^{-N} .
If f_{p} satisfies Ñ_{p}^{2} f_{p} = lf_{p}
(such as f(x)=f(|x|) where f"(r) +(N-1)r^{-1}f'(r) = ±lf(r)
according as x^{2} is ±)
then f_{p}d solves the Laplace Equation for any p-independant d.
If f(r) satisfies f"(r) + (N-1)r^{-1}f'(r) - (N-1)r^{-2}f(r)
= lf(r) then f(|x|)x^{~} satisfies Ñ_{x}^{2} f(|x|)x^{~} =
l f(|x|)x^{~} .
[ Proof :
Ñ(f(|x|)x^{~}) = (Ñf(|x)|)x^{~} + f(|x))Ñ(x^{~})
= ±f'(r)x^{~}^{2} + f(r)(N-1)r^{-1}
= f'(r) + f(r)(N-1)r^{-1}
Ñ_{x}^{2}f(r)x^{~}
= ±f"(r)x^{~} ±f'(r)(N-1)r^{-1}x^{~}
-/+ f(r)(N-1)r^{-2}x^{~}
= ±(f"(r) + (N-1)r^{-1}f'(r)-(N-1)r^{-2}f(r))x^{~}
.]
Thus we seek f(r) solving
f"(r) + (N-1)r^{-1}f'(r) + (l - (N-1)r^{-2})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}.p' the existance of a
pervading Faraday electromagnetic bivector force field quantifiable (along p(t)) as
f_{p(t)} = (m_{0}/q_{0})W(t) + b_{2}(p(t))
where arbitary spacelike 2-blade b_{2} Î p'^{*} cannot be quantified (deduced) from the trajectory ;
and proper mass m_{0} and proper charge q_{0} 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 p'(t) . It does not arise in Â_{1,3} timespace.
If we instead use p" = (W_{p}p')_{<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-R^{2}w^{2})^{-½}t e_{21})^{↑}_{§}(e_{3}) we have p"(t)p'(t) = -(1-R^{2}w^{2})^{-3/2}w^{2}R R_{t}_{§}(Rwe_{21} + e_{41}) so (p"(t)p'(t))^{2} = (1-R^{2}w^{2})^{-2}w^{4}R^{2} .
Bivector field f_{p} decomposeses into electrostatic and magnetic components = c^{-1}e_{p} + b_{p} where b_{p} = Be_{123}^{-1} lies in Îe_{123} and e=Ee_{4} has e_{4}Ùe_{p}=0
We "incorporate the local medium" by defining electric displacement
D_{p} = e_{p}E_{p}
and
magnetic intensity spacial bivector h_{p} = m_{p}^{-1}b_{p}
where linear dilectrics e_{p} and m_{p}^{-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 e_{p} = m_{p} = 1
in a "vacuum". In S.I. units we have
e_{0}
= (2hca)^{-1}Q_{e-}^{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} s^{4} A^{2}
= 1 m^{-2} kg^{-1} s^{2} C^{2}
= 10^{-7}c^{2} m ;
and
m_{0} = 4p×10^{-7} H m^{-1} = 4p where
one henri = 1 H = 1 m^{2} kg s^{-2} A^{-2}
= 1 m^{2} kg C^{-2} = 10^{7} m .
Bivector field g_{p} =
cD_{p}e_{4} - H_{p}e_{123}
= ce_{p}e_{p} - m_{p}^{-1}b_{p} provides a geometric analog of the Minkowski tensor form of the Maxwell
equations
Ñ_{p}¿g_{p} = j_{p} ;
Ñ_{p}Ùf_{p} = 0
which, if
g_{p} = m^{-1}f_{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_{p}f_{p} = Ñ_{p}¿f_{p} =
m j_{p} .
Splitting the "dilectric" into e_{4}-relative scaling components feels nonrelativistic, and it is more natural
to replace
g_{p} = ce_{p}e_{p} - m_{p}^{-1}b_{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 e_{4}^{*}
]
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}=0 as an
inevitable consequence of its construction.
If the 1-potential a_{p} satisies Ñ_{p}¿a_{p}=0 then we have Ñ_{p}a_{p}=-f_{p}
and hence
j_{p} = m_{p}^{-1} Ñ_{p}(f_{p}) = -m_{p}^{-1} Ñ_{p}^{2} a_{p}
= -m_{p}^{-1} Ñ_{p}¿(Ñ_{p}Ùa_{p})
with m_{p}^{-1}=m^{-1} a point-independant scalar in an isotrophic linear medium.
.
[ In Â_{1,3} we have Ñ_{p}^{2}a_{p} = mj_{p} ]
In a classical vacuum j_{p}=0 and so Ñ_{p}f_{p} = 0 which has pure
Â_{3,1} bivector
circularly polarised wave solution
f_{p} = ak((k_{*}p)e_{1234})^{↑}
for any null 1-vector k and 1-vector a normal
to k where a^{↑} º e^{a} denotes exponentiation. 4-blade e_{1234} is available as a commuting i here because
f_{p} is even.
Picking a frame with k=w(e_{3}+e_{4}) we observe that ke_{34} = k so that
k((k_{*}p)e_{1234})^{↑} = k((k_{*}p)e_{12})^{↑} and we can express our solution as
f_{p} = a(e_{3}+e_{4})((w(e_{3}+e_{4})_{*}p)e_{12})^{↑}
= re_{1}(e_{3}+e_{4})((q+(w(e_{3}+e_{4})_{*}p))e_{12})^{↑}
= re_{1}(e_{3}+e_{4})((q+(w(e_{3}+e_{4})_{*}p))e_{1234})^{↑}
where a = re_{1}e^{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)^{a}b)^{↑} kA
has Ñ_{p}f_{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_{p}F_{p} = lF_{p}
that commutes with null 1-vector k then F_{p}k((p¿k)^{-2j})^{↑}
for integer j³1
is also solves Ñ_{p}F_{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)^{-2j}b)^{↑} 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))e_{4} - B_{E}(p(t))e_{123}
where e_{123} 1-vectors
E_{E}(p(t)) º ^_{e4(t)*}(f(p(t)))e_{4}^{-1}
and B_{E}(p(t)) º - ¯_{e4(t)*}(f(p(t))) e_{123}^{-1}
are known as the relative electric field
and the relative magnetic field respectively.
Relative magnetic fields are more naturally represented by e_{4}^{*} bivector
b_{p} º b_{E}(p(t)) = B_{p}e_{123} giving
f_{p} = c^{-1}e_{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")
= (q_{0}/m_{0}) ¯(f,e_{4}^{*})
= (q_{0}/m_{0})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[t_{0},t_{1}]} .
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 (t_{0} 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[t_{0},t_{1}].
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_{p}^{2}=-h_{0}^{2}
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_{p}t_{p}) = 0 ; and v_{p}¿(Ñ_{p}t_{p}) = 1 .
[ Proof :
Ñ_{p}(s_{p}_{Ñ}¿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_{p}t(p))v_{p}¿s_{p} - s_{p}
But Ñ_{p}(s_{p}^{2})=Ñ_{p}(h_{0}^{2})=0 so (Ñ_{p}t(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_{p}t_{p})v_{p} - N = (s_{p}¿v_{p})^{-1} s_{p}v_{p} - N
(s_{p}¿v_{p})^{-1} s_{p}v_{p} - N .
Ñ_{p} v_{p} = (Ñ_{p}t_{p})v_{p}' = (s_{p}¿v_{p})^{-1} s_{p}v_{p}' .
Ñ_{p} (s_{p}¿v_{p})
= -(s_{p}¿v_{p})^{-1} ( s_{p}(s_{p}Ùv_{p}') + v_{p}(s_{p}Ùv_{p}))
= -½(s_{p}¿v_{p})^{-1} ( v_{p}s_{p}v_{p} + s_{p}^{2}v_{p}' - v_{p}^{2}s_{p} - s_{p}v_{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_{p}_{Ñ}
+ (v_{p}¿Ñ)s_{p}_{Ñ}
- s_{p}¿(Ñ_{p}Ùv_{p}) - v_{p}¿(Ñ_{p}Ùs_{p})
= Ð_{sp}v_{p} + Ð_{vp}s_{p} - s_{p}¿(Ñ_{p}Ùv_{p}) - v_{p}¿(Ñ_{p}Ùs_{p}) .
The directed scalar-funnelled rule
Ð_{A}F(g(X)) = F'(g(X))g^{Ñ}(A)=(A_{*}(Ñ_{x}g(X))) F'(g(x)) enables verification that
Ð_{vp}s_{p} = Ð_{sp}v_{p} = 0 so we have
Ñ_{p}(s_{p}¿v_{p})
= - s_{p}¿(Ñ_{p}Ùv_{p}) - v_{p}¿(Ñ_{p}Ùs_{p})
= - s_{p}¿(Ñ_{p}v_{p})_{<2>}
- v_{p}¿(Ñ_{p}s_{p})_{<2>}
= -(s_{p}¿v_{p})^{-1} ( s_{p}¿(s_{p}Ùv_{p}') + v_{p}¿(s_{p}Ùv_{p}))
= -(s_{p}¿v_{p})^{-1} ( s_{p}(s_{p}Ùv_{p}') + v_{p}(s_{p}Ùv_{p}))
= -(s_{p}¿v_{p})^{-1} ½( s_{p}(s_{p}v_{p}'-v_{p}'s_{p}) + v_{p}(s_{p}v_{p}-v_{p}s_{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_{p}Ùv_{p}') vanishes.
If s_{p}=-g_{U}(e_{4}+U_{p}) is backward timelike, s_{p}¿v_{p} is minimised
by timelike v_{p} =g_{V}(e_{4}+V_{p}) as g_{U}g_{V}(1-V_{p}^{2}U_{p}^{2}) where U_{p} 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®Â.
We have
f_{p}
= -Ñ_{p} a_{p}
=
-(s_{p}¿v_{p})^{-1} (
f'(s_{p}¿v_{p})( -s_{p}(s_{p}Ùv_{p}')v_{p} + v_{p}^{2}(s_{p}Ùv_{p}))
+ f(s_{p}¿v_{p})s_{p}v_{p}'
)
[ Proof :
Ñ_{p}(f(s_{p}¿v_{p})v_{p})
= (Ñ_{p}f(s_{p}¿v_{p}))v_{p} + f(s_{p}¿v_{p}) (Ñ_{p}v_{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_{p}v_{p}'
= -f'(s_{p}¿v_{p})(s_{p}¿v_{p})^{-1} (s_{p}(s_{p}Ùv_{p}')+v_{p}(s_{p}Ùv_{p}))v_{p}
+ f(s_{p}¿v_{p})(s_{p}¿v_{p})^{-1} s_{p}v_{p}'
= (s_{p}¿v_{p})^{-1}(
f'(s_{p}¿v_{p})( -s_{p}(s_{p}Ùv_{p}')v_{p}+v_{p}^{2}(s_{p}Ùv_{p}))
+ f(s_{p}¿v_{p})s_{p}v_{p}') .]
Setting f(s_{p}¿v_{p}) = a(h_{0}-(s_{p}¿v_{p})^{2})^{-½}
for some p-independant fixed scalar "mass" or "charge" or "ammount" a we have
f_{p} = -Ñ_{p}a_{p}
= -(s_{p}¿v_{p})^{-1} (
-½((h_{0}-(s_{p}¿v_{p})^{2})^{-3/2}
(s_{p}¿v_{p})( -s_{p}(s_{p}Ùv_{p}')v_{p} + v_{p}^{2}(s_{p}Ùv_{p}))
+ (h_{0}-(s_{p}¿v_{p})^{2})^{-½}
s_{p}v_{p}'
)
= -(s_{p}¿v_{p})^{-1} (h_{0}-(s_{p}¿v_{p})^{2})^{-3/2}(
-½(
(s_{p}¿v_{p})( -s_{p}(s_{p}Ùv_{p}')v_{p} + v_{p}^{2}(s_{p}Ùv_{p}))
+ (h_{0}-(s_{p}¿v_{p})^{2})s_{p}v_{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_{p}^{2}(s_{p}Ùv_{p})
+ (s_{p}v_{p}')×(s_{p}v_{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_{p}v_{p})
= -(s_{p}¿v_{p})^{-2}(Ñ_{p}(s_{p}¿v_{p}))v_{p} + (s_{p}¿v_{p})^{-2} s_{p}v_{p}'
= (s_{p}¿v_{p})^{-3}(½( v_{p}s_{p}v_{p}
+ s_{p}^{2}v_{p}' - v_{p}^{2}s_{p} -s_{p}v_{p}'s_{p})v_{p}
+ (s_{p}¿v_{p}) s_{p}v_{p}')
= ½(s_{p}¿v_{p})^{-3}( v_{p}^{2}v_{p}s_{p}
+ s_{p}^{2}v_{p}'v_{p} - v_{p}^{2}s_{p}v_{p} - s_{p}v_{p}'s_{p}v_{p}
+ (s_{p}v_{p}+v_{p}s_{p})s_{p}v_{p}')
= (s_{p}¿v_{p})^{-3}( s_{p}^{2}(v_{p}¿v_{p}')
+ v_{p}^{2}(v_{p}Ùs_{p})
+ (s_{p}v_{p})×(s_{p}v_{p}')
)
= (s_{p}¿v_{p})^{-3}(
v_{p}^{2}(v_{p}Ùs_{p})
+ (s_{p}v_{p})×(s_{p}v_{p}')
)
.]
Both terms of Ñ_{p}a_{p} are pure bivectors so Ñ_{p}¿a_{p} = 0, as required of a 1-potential,
for any trajectory q(t). Note that
(s_{p}v_{p}')×(s_{p}v_{p})
= (s_{p}v_{p}')(s_{p}v_{p}) - (s_{p}v_{p}')¿(s_{p}v_{p})
= (s_{p}v_{p}'s_{p})v_{p} - (s_{p}v_{p}'s_{p})¿v_{p}
= (s_{p}v_{p}'s_{p})Ùv_{p}
= -(s_{p}v_{p}s_{p})Ùv_{p}'
When n^{2}=-1 we have -nxn = ¯_{n}(x) - ^_{n}(x) so
(s_{p}v_{p}')×(s_{p}v_{p})
= -((¯_{sp}(v_{p}')-^_{sp}(v_{p}'))Ùv_{p}
= ((¯_{sp}(v_{p})-^_{sp}(v_{p}))Ùv_{p}' .
= (s_{p}v_{p}')(s_{p}v_{p}) - (s_{p}v_{p}'s_{p}v_{p})_{<0>}
= (s_{p}v_{p}')(s_{p}v_{p}) + ((¯(v_{p}',s_{p})-^(v_{p}',s_{p}))v_{p})_{<0>}
= (s_{p}v_{p}')(s_{p}v_{p}) + (¯(v_{p}',s_{p})-^(v_{p}',s_{p}))¿v_{p}
= (s_{p}v_{p}')(s_{p}v_{p}) + (¯(v_{p},s_{p})-^(v_{p},s_{p}))¿v_{p}' .
Since f_{p} acts via u_{p}' = (qm^{-1}) f_{p}¿u_{p} we can add a scalar part to f_{p} provided we
make the ¿ in Maxwells equation Ñ_{p}f_{p} = Ñ_{p}¿f_{p} = m_{p} j_{p}
explicit.
It is thus natural to set <0;2>-field
¦_{p} = a (s_{p}¿v_{p})^{-3}(
v_{p}^{2}s_{p}v_{p})
+ (s_{p}v_{p}')(s_{p}v_{p})
)
= a (s_{p}¿v_{p})^{-3}s_{p}v_{p}'s_{p}v_{p}
for null v_{p}.
For null s_{p}, the (s_{p}v_{p}')×(s_{p}v_{p}) = (s_{p}Ùv_{p}')×(s_{p}Ùv_{p}) term is equal to
-2s_{p}(v_{p}'Ùv_{p})s_{p} =
-2s_{p}W_{p}s_{p} = -4(s_{p}¿W_{p})s_{p} = -4(s_{p}¿W_{p})Ùs_{p}
where W_{p}=-v_{p}'Ùv_{p}.
[ by the
null reflected 2-blade rule
sabs = s(aÙb)s = sbsa - sasb
]
and we obtain
f_{p}
= av_{p}^{2}(s_{p}¿v_{p})^{-2} (s_{p}Ùv_{p})^{~}
+ ½a(s_{p}¿v_{p})^{-3} s_{p}W_{p}s_{p}
for the null-propagated Lienard-Wiechart field.
If v_{p} is also null we have the purely radiative
f_{p}
=
½a(s_{p}¿v_{p})^{-3} s_{p}W_{p}s_{p}
.
Thus a charged nonaccelerating null-trajectory particle emmits no field. Acceleration (v_{p}'¹0
generates a tightly pulsed r^{-1} field that tends to alter orbital momentum about the wordline
(Biot-Savart law) . Timelike traversal incurrs an r^{-2} s_{p}Ùv_{p} field that
never vanishes but approches 0 as (s_{p}¿v_{p}) ® ¥ .
In the electromagnetic ("current") case, we have a = (4pe_{0})^{-1}q
where q is the "fixed charge" of the particle. The Faraday bivector at p is given by
-Ñ_{p}Ùa_{p} = -Ñ_{p}a_{p} and so contains an "invere square" Coloumb boost ,
central for a static (v_{p}=e_{4}) wordline ;
and a mostly tangential "reciprocal law" radiation ( Bremstrahlung, synchrotronic)
null boost due to the particle acceleration v_{p}'.
In the gravitic (momentum) case these are regarded as an inverse square "Newton" boost and
a reciprocal "gravitomagnetic" boost.
For a null worldline with v_{p}^{2}=0 generating a null Lienard-Wiechart 1-potential,
the inverse square term vanishes and we are left only with the radiation term.
The unbounded |v_{p}¿s_{p}|^{-3} factor is infinite only when
(v_{p}'Ùv_{p})s_{p}=0 so such a null 4-potential is physically plausable.
If we allow a=a_{p}=a(t(p)) to vary with t we aquire an additional (Ña_{p}) (s_{p}¿v_{p})^{-1} v_{p} term in Ñ_{p}a_{p} which is expressable as (a'_{p}) (s_{p}¿v_{p})^{-2} s_{p}v_{p} and is not a pure bivector since s_{p}v_{p} has a scalar part. Thus if a partcle varies its charge or mass, the resultant 1-potential has nonzero point divergence.
When s_{p}^{2}=h_{0}^{2} ; v_{p}^{2}=0
we have f_{p}
= -Ñ_{p} a_{p}
= a (s_{p}¿v_{p})^{-3} (s_{p}v_{p}')×(s_{p}v_{p})
= a (s_{p}¿v_{p})^{-3} ((s_{p}Ùv_{p}')(s_{p}Ùv_{p}))_{<2>}
and can set
u_{p}' = (s_{p}¿v_{p})^{-3} ((s_{p}Ùv_{p}')(s_{p}Ùv_{p})u_{p})_{<1>}
.
Field due to straight worldine
When v_{p}'=0 we have f_{p} = a (s_{p}¿v_{p})^{-3}
v_{p}^{2}(s_{p}Ùv_{p})
which is 0 for a v_{p}^{2}=0 null worldine and
-a (s_{p}¿e_{4})^{-3}(s_{p}Ùe_{4}) for the v_{p}^{2}=-1 timelike worldine
{te_{4} : t Î [-¥,¥] } .
For null propagation the field at p=X+te_{4}
for XÎe_{123} is the
the inverse square Coulomb field
((X-|X|e_{4})¿e_{4})^{-3}(X-|X|e_{4})Ùe_{4}
= |X|^{-2}(Xe_{4})^{~} .
For s_{p}^{2} = ±h_{0}^{2} and |X|^{2} > h_{0}^{2} we have
s_{p}=X-(X^{2} -/+ h_{0}^{2})^{½}e_{4}
and field
-a(X^{2} -/+ h_{0}^{2})^{-3/2}XÙe_{4}
closely approximating the Coloumb field for |X| >> h_{0}.
Instananeous Field due to helical worldine
Consider a particle orbitting 0 at distance R within the e_{12} plane having proper time formulation
q(t) =
(½wgt e_{21})^{↑}_{§} (
Re_{1} + tge_{4})
where
g = |1-R^{2}w^{2}|^{-½}
and |Rw| < 1.
We may be particularly interested in R = ¼hp^{-1} m^{-1}c^{-1}
(ie. ½m^{-1} in natural units).
At t=0 we have q'(0)= g(Rwe_{2} + e_{4}) ; q"(0) = -(1-R^{2}w^{2})^{-1}w^{2}Re_{1} ; and q'(0)q"(0) = -g^{3}w^{2}R(Rwe_{21} + e_{41}) .
The interesction of a helix and a null cone is algebraically complex but can be numerically established,
eg. via a binary search on t, relatively straightforwardly.
The e_{4}-instantaneous field takes the form of a fixed width and span spiral containing a pulse of strength in proportion to r^{-1} .
If we integrate the field over one period at a fixed spacial location P (from P to P + 2pw^{-1}e_{4} say)
then, apart from in a localised "ring" region with r»1 , the e_{12} and
e_{q}e_{4} componensts average to zero while the radial
e_{r}e_{4} components averages to uniform radial field proportionate to r^{-2}.
Such integrations are both computationally problematic (the spiral wavefront is travelling at
at supralight speeds for r>1 and "sweeps over" a test point very rapidly, making the integration
one of a quasi-discontinuous Dirac-delta like spike) but also physically suspicous since a test partcle will actually move with the field to some degree,
The radial
e_{r}Ùe_{q} pulse, for example, will ted to slightly increase r
via a tangental kick, and so the "restoring" effect will be reduced due to the r^{-1} factor.
Retarded Lienard-Wiechart Field due to unit radius helix at e_{4}-time t=0
when particle is at e_{1}. 10 × 10 square area
radially scaled by r over r>1 ; logged and clipped at bright red (negative) and bright green (positive).
Bottom left quadrant shows e_{r}Ùe_{4} "Coulomb" component , bottom right shows
e_{q}Ùe_{4} "Biot-Savart" compoent , top left shows
e_{r}Ùe_{q}=e_{12} "magnetric moment" component.
Top right shows (nonphysical?) scalar part.
Note how field apparently changes sign due to the narrowing of the central peak, and contracts slightly as g increases from 1¼ (slow) to ¥ (lightspeed). | ||||
g=5/4 10×10 region | g=3/2 10×10 region | g=2 10×10 region | ||
g=2 1/4 10×10 region | g=3 10×10 region | g=¥ 10×10 region | ||
g=2 2^{7} sample average | ||||
g=4 2^{7} sample average | ||||
g=2 20×20 region | g=¥ 20×20 region | T-averaged g=2 field. Fine structure is computional artifact from sampling timestep 2^{-7}T ie. at roughly the Treiman timestep. | ||
g=2 20×20 region | g=¥ 20×20 region | 20×20 g=¥ timelike seperation | ||
We now plot spacelike unit helices (R=1, w > 1), Retaining 10×10 area we scale all but the e_{r4} field by w^{-1} . | ||||
w=2^{-1}p | w=p | w=2p | ||
w=2^{2}p | w=2^{4}p |
The spin in the trajectory thus introduces an oscillation in the field. Rather than being smmothly varying over time and space,
the field is "bunched" into a pulse with a profile that changes sign. The pulse is preceeded by an opposite torque that serves
to draw a test charge into a repulsive wave and away from an attractive one. This will tend to amplify repulsive effects
compared to attractive ones.
Averaged Field due to helical worldine
Integrating the field at spaical point P over the helical period (2p)^{-1}w gives the net imupulse applied by a single
field pulse to a test charge fixed at P. This is somewhat unphysical since a free test charge that "rides" the field
and moves slightly during the pulsetime may experience a significantly different impulse due to being
"brushed aside from" or "drawn in to" the main pulse. Nonetheless, if we think of a compartively heavy test charge
slow to respond to the field, summing the field at P over e^{4}¿p Î[t,t+(2p)^{-1}w]
provides a reasonable indicator of the effective field at p.
Analytically, such integerations are nontrivial. Computationally, they are problematic in
that the integrated field is exceptionally bunched making naive applications of Simpson's Rule
likely to "miss" the pulse and give high errors. Adpative refinement methods are required and, since
each field evaluation involves a worldline intersection, the calculation load is significant.
Self Interaction
Timelike self interaction of null helix
For timelike s_{p}^{2}=-h_{0}^{2} we have the possibility of self-interaction. Recalling that v_{p} and W_{p}
describe the matter at p+s_{p}, and letting u_{p} and K_{p} denote the matter at p
we have u_{p}'=u_{p}¿K_{p} = u_{p}¿((qm^{-1})f_{p})
= (qm^{-1}) u_{p}¿f_{p}
where
K_{p} =-u_{p}'u_{p} = g^{3}w^{2}R R_{t}_{§}(Rwe_{12} + e_{14})
= -g^{3}w^{3}R^{2}e_{12} - g^{3}w^{2}R R_{t}_{§}(e_{14}) .
Suppose p=Re_{1} and s_{p}
= -2kpw^{-1}e_{4} with s_{p}^{2} =
-4(kp)^{2}w^{-2} so that
v_{p} = u_{p} = gRwe_{2} + ge_{4}
;
v_{p}' = u_{p}' = -g^{2}w^{2}Re_{1} .
It follows that, at p=Re_{1}, f_{p} = a (s_{p}¿v_{p})^{-3}(
v_{p}^{2}(s_{p}Ùv_{p}) + (s_{p}Ùv_{p}')×(s_{p}Ùv_{p}) )
= a
(2kpw^{-1}g)^{-3}
(
-2kpgRe_{24}
-4(kpg)^{2}wR^{2}ge_{12} )
[ Proof :
s_{p}¿v_{p} =
(-2kpw^{-1}e_{4})¿(gRwe_{2} + ge_{4})
= 2kpw^{-1}g
s_{p}Ùv_{p}
= 2kpgRe_{24} ;
s_{p}Ùv_{p}' = (-2kpw^{-1}e_{4})Ù(-g^{2}w^{2}Re_{1})
= -2kpwRg^{2}e_{14} .
(s_{p}Ùv_{p}')×(s_{p}Ùv_{p})
= -2kpwRg^{2}e_{14} × 2kpgRe_{24}
= -4(kpg)^{2}wR^{2}ge_{12}
and result follows.
.]
When v_{p}^{2}=0 (so Rw=g=1) we have f_{p}
= - a(2kpw^{-1})^{-3} 4(kp)^{2}R e_{12}
= -a(2kp)^{-1}w^{2} e_{12}
and so
(qm^{-1})f_{p}¿u_{p}
= -(qm^{-1})a(2kp)^{-1}w^{2} e_{1}
where m is the "inertial mass" of the worldline.
This is
equal to the desired u_{p}' = -we_{1} if
(qm^{-1})a(2kp)^{-1}w = 1 , ie. if
m = (2kp)^{-1}aq w
= (2kp)^{-1}aq (2ph_{0}^{-1})
= k^{-1}aq h_{0}^{-1}
.
It is conventional to set a = q(4pe_{0})^{-1}
giving
m
= k^{-1}q^{2} (4pe_{0})^{-1} h_{0}^{-1}
= a k^{-1} h_{0}^{-1} where a
= q^{2} (4pe_{0})^{-1}
[ q^{2} (4pe_{0} (2p)^{-1}hc)^{-1}
in unnatural units ]
is the fine structure constant.
Hence choosing a particluar effectation seperation h_{0} we have a selfinteracting null helical equilibrium with R=(2p)^{-1}h_{0} , w = ± 2ph_{0}^{-1} , and period h_{0} provided the inertial mass attributed to the particle is m = k^{-1}a h_{0}^{-1} for some integer windage number k , the worldine being influenced by itself k periods back .
Setting q=1 , our model is thus a null worldine radiating potential a_{p} = a (s_{p}¿v_{p})^{-1} v_{p} = (4pe_{0})^{-1} (s_{p}¿v_{p})^{-1} v_{p} and responding to the induced field f_{p} as u_{p}' = (qm^{-1}) f_{p} ¿ u_{p} = k(4pe_{0})h_{0} f_{p} ¿ u_{p}. = -kh_{0} (Ñ_{p}((s_{p}¿v_{p})^{-1}v_{p})) ¿ u_{p}.
If R >> 2ph_{0} there will be some time t such that
s_{p} = (1-R cos(wt))e_{1} - R sin(wt)e_{2} - te_{4}
has
s_{p}^{2} =((½wt)-1)^{2} - t^{2} =
R^{2}2(1- cos(wt)) - t^{2} = -h_{0}^{2}
and |s_{p}v_{p}| will tend to be small so f_{p} will be large and tend to increase the acceleration, decreasing
the radius of the helix. Once a null streamline deviates from inertial, perhaps by the influence of a seperate worldine,
then it will "capture its own past" and coil into a tight self-maintaining helix.
.
We then have
s_{p}¿v_{p} =
( (1-R cos(wt))e_{1} - R sin(wt)e_{2} - te_{4})¿(e_{2}+e_{4})
= - R sin(wt) + t
;
s_{p}Ùv_{p} =
( (1-R cos(wt))e_{1} - R sin(wt)e_{2} - te_{4})Ù(e_{2}+e_{4})
= (1-R cos(wt))e_{1}Ù(e_{2}+e_{4}) + (t-R sin(wt))e_{24} ;
and s_{p}Ùv_{p}' =
-( (1-R cos(wt))e_{1} - R sin(wt)e_{2} - te_{4}) Ù _ome_{1}
=
- sin(wt))e_{12} - te_{14} .
(s_{p}Ùv_{p}')×(s_{p}Ùv_{p}) = - ( sin(wt))e_{12} + te_{14}) × ((1-R cos(wt))e_{1}Ù(e_{2}+e_{4}) + (t-R sin(wt))e_{24}) ; -
Field due to unit radius null and timelike propagated null helix at e_{4}-time t=0 when particile is at e_{1}. 10×10 region. | ||||
g=¥ null seperation
as above but brighter colouring | g=¥ timelike seperation
Weaker and wider field | g=¥ timelike seperation with increased brightness. |
More generally a helix will "experience itself" at p(0)=Re_{1} as the field it radiated at p(-t) where t solves (p(t)-Re_{1})^{2}=h_{0}^{2} Û R^{2}((1- cos(½wgt))^{2} + sin(½wgt)^{2}) - g^{2}t^{2} = h_{0}^{2} Û 2R(1- cos(wgt)) - g^{2}t^{2} = h_{0}^{2} Û 2R(1- cos(wgt)) - g^{2}t^{2} = h_{0}^{2} Û 2R - (¼w^{2}R + 1)g^{2}t^{2} - 2Rå_{j=2}^{¥} (wgt)^{2j}(2j)!^{-1} = h_{0}^{2} .
More generally still, we have the field at Re_{1} dues to a helix p(t)= R((wgt+f)e_{12})^{↑}e_{1} + gte_{4}
where f is a phase angle, providing self interaction when f=0.
Null self interaction of spacelike helix
For a null-seperated self-interaction with s_{p}^{2}=0 we require a spacelike helix with v_{p}^{2}=1. Suppose p=Re_{1} and s_{p} = -2Re_{1} - kpw^{-1}e_{4} with s_{p}^{2} = 4R^{2}-k^{2}p^{2}w^{-2} . For s_{p}^{2}=0 we require Rw = ± ½kp .
Set g = g_{k} º ((½kp)^{2} - 1)^{-½} and
h_{k} º wRg_{k} = ±(½kp)((½kp)^{2} - 1)^{-½}
with h_{k}^{2}-g_{k}^{2} = 1.
v_{p}
= -gRwe_{2} + ge_{4} = -h_{k}e_{2} + g_{k}e_{4}
;
v_{p}' = g^{2}w^{2}Re_{1} = R^{-1}h_{k}^{2}e_{1}
u_{p} = gRwe_{2} + ge_{4} = h_{k}e_{2} + g_{k}e_{4}
; u_{p}' = -g^{2}w^{2}Re_{1}
= -R^{-1}h_{k}^{2}e_{1} .
s_{p}¿v_{p} =
(-2Re_{1} - kpw^{-1}e_{4})¿(-gRwe_{2} + ge_{4})
= kpw^{-1}g
s_{p}Ùv_{p} =
(-2Re_{1} - kpw^{-1}e_{4})Ù(-gRwe_{2} + ge_{4})
= 2R^{2}gwe_{12} - kpgRe_{24} - 2Rge_{14}
= Rg(2Rwe_{12} - kpe_{24} - 2e_{14})
s_{p}Ùv_{p}' = (-2Re_{1} - kpw^{-1}e_{4})Ù g^{2}w^{2}Re_{1}
= kpwRg^{2}e_{14} .
(s_{p}Ùv_{p}')×(s_{p}Ùv_{p})
= kpwRg^{2}e_{14}× Rg(2Rwe_{12} - kpe_{24})
= kpwR^{2}g^{3}(2Rwe_{24} - kpe_{12})
These yield f_{p} = a (s_{p}¿v_{p})^{-3}(
v_{p}^{2}(s_{p}Ùv_{p})
+ (s_{p}Ùv_{p}')×(s_{p}Ùv_{p}) )
= -/+ a ½(kp)^{-2}w^{2}g_{k}
( 2e_{14} + (kp)(1-2h_{k}^{2})(-e_{12}±e_{24})
)
[ Proof :
a (kpw^{-1}g)^{-3}
( Rg(2Rwe_{12} - kpe_{24} - 2e_{14})
+ kpwR^{2}g^{3}(2Rwe_{24} - kpe_{12}))
= a (kp)^{-3}g^{-2}w^{3} R
(2Rwe_{12} - kpe_{24} - 2e_{14}
+ kpwRg^{2}(2Rwe_{24} - kpe_{12}))
= ± a ½(kp)^{-2}w^{2}g^{-2}
(±kpe_{12} - kpe_{24} - 2e_{14}
± ½(kp)^{2}g^{2}(±kpe_{24} - kpe_{12}))
= a ½(kp)^{-2}w^{2}g^{-2}
(kpe_{12} -/+ (kpe_{24} + 2e_{14})
+ ½(kp)^{2}g^{2}(±kpe_{24} - kpe_{12}))
= a ½(kp)^{-2}w^{2}g^{-2}
( -/+ 2e_{14} + (kp)
( (1-½(kp)^{2}g^{2})e_{12}
+ (±½(kp)^{2}g^{2} -/+ 1) e_{24}
)
= a ½(kp)^{-2}w^{2}g^{-2}
( -/+ 2e_{14} + (kp)(1-½(kp)^{2}g^{2})(e_{12} -/+ e_{24})
)
= a ½(kp)^{-2}w^{2}g^{-2}
( -/+ 2e_{14} - (kp)(1-2h_{k}^{2})(-e_{12} ± e_{24})
)
= ± a ½(kp)^{-2}w^{2}g^{-2}
( -2e_{14} - (kp)(1-2h_{k}^{2})(-e_{12} ± e_{24})
)
since
(1-½(kp)^{2}g^{2}) = (1-½(kp)^{2}((½kp)^{2}-1) .
.]
Hence f_{p}¿u_{p}
= -/+ a ½(kp)^{-2}w^{2}g_{k}
( 2e_{14} + (kp)(1-2h_{k}^{2})(-e_{12}±e_{24})
)
¿ (h_{k}e_{2} + g_{k}e_{4})
= -/+ a ½(kp)^{-2}w^{2}g_{k}
( -2g_{k}e_{1}
+ (kp)(1-2h_{k}^{2})(-h_{k}e_{1}-/+h_{k}e_{4} -/+ge_{2})
)
= ± a ½(kp)^{-2}w^{2}g_{k}
( 2g_{k}e_{1}
+ (kp)(1-2h_{k}^{2})(h_{k}e_{1}±h_{k}e_{4} ±ge_{2})
)
The e_{1} coordinate of
u_{p}' = (qm^{-1})f_{p}¿u_{p}
is thus
± (qm^{-1})a ½(kp)^{-2}w^{2}g_{k}
( 2g_{k} + (kp)(1-2h_{k}^{2})h_{k}) which equals the desired
-g^{2}w^{2}R if
= a ½(kp)^{-2}w^{2}g
( -/+ 2e_{14} - (kp)(1-2h_{k}^{2})(e_{12} ± e_{24}))
¿ (±h_{k}e_{2} + g_{k}e_{4})
= a ½(kp)^{-2}w^{2}g
( (±2 -/+ (kp)(1-2h_{k}^{2})h_{k})e_{1}
± (kp)(1-2h_{k}^{2})(g_{k}e_{2}+h_{k}e_{4}) )
Hence f_{p}¿u_{p}
= a (kpw^{-1}g)^{-3} Rg
( Rw(2-(kpg)^{2})e_{12}
+ kp(2w^{2}R^{2}g^{2}-1)e_{24}
- 2e_{14}) ¿ g(Rwe_{2} + e_{4} )
= a (kp)^{-3}w^{3}Rg^{-1}
( (2+(Rw)^{2}(2-(kpg)^{2})e_{1}
+ kpRw(1-2w^{2}R^{2}g^{2})e_{2}
+ kpRw(1-2w^{2}R^{2}g^{2})e_{4}
)
= a (kp)^{-3}w^{3}Rg^{-1}
( (2+(Rw)^{2}(2-(kpg)^{2})e_{1}
- kpRw(e_{2}+e_{4})
)
[ since (1-2w^{2}R^{2}g^{2}) = g^{2}(g^{-2}-2w^{2}R^{2})
= -g^{2}(w^{2}R^{2}-1) = -1 when |Rw| > 1 ]
Setting Rw = ± ½kp ,
g = g_{k} = (2^{-2}k^{2}p^{2}-1)^{-½}
so that (2-(kpg)^{2}) = ((½kp)^{2}-1)^{-1}(2(½kp)^{2}-3)
gives
f_{p}¿u_{p} =
a (kp)^{-3}w^{2} (±½kp)g_{k}^{-1}
( (2+¼(kp)^{2}(2-(kpg)^{2})e_{1}
-/+ ½(kp)^{2}(e_{2}+e_{4})
)
a w^{2} (±½)g_{k}^{-1}
( (2(kp)^{-2}+¼(2-(kpg)^{2})e_{1}
-/+ ½(e_{2}+e_{4})
)
.
The e_{1} coordinate of u_{p}' = (qm^{-1})f_{p}¿u_{p}
is thus
(qm^{-1}) a w^{2} (±½)g_{k}^{-1}
( 2(kp)^{-2}+¼(2-(kpg)^{2}))
which equals the desired
-g^{2}w^{2}R if
g^{2}R
= (qm^{-1}) a ½g_{k}^{-1} ( 2(kp)^{-2}+¼(2-(kpg)^{2}))
Û R = (qm^{-1}) a g_{k}^{3}( ¼ + (kp)^{-2} - 1/8(kpg)^{2})
.
Constructing the potential
The previous discussion constructed the potential at a given event p due to a worldline q(t)
in accordance with null-seperated interactions. We considered only the "particle-instant" p+s_{p} in the
rear null cone L^{-}_{p} to contribute, perhaps by means of a virtual particle traversing a straight null path.
Becuase we considered the matter at p+s_{p} to be represented by the timelike v(t(p))=v_{p} it was natural to allow it
to contribute a v_{p}-static potential of zero-divergent magnitude (s_{p}¿v_{p})^{-1} extended out from
p+s_{p} with v_{p} considered constant. The higher derivatives like v'(t)=p"(t) did not directly contribute to the
"instantaneous" potentiala at p, effecting it only indirectly by influencing s_{p}
Given a_{p0} = a (s_{p}¿v_{p})^{-1} v_{p}
= a (s_{p}¿j_{p})^{-1} j_{p}
= -a (¯_{vp}(s_{p}))^{-1} , it
is natural to consider a small 4-simplex d^{4}p at pÎ L^{-}_{p0}
as generating a potential
a |d^{4}pj_{p}|((p-p_{0})¿v_{p})^{-1} v_{p}
= a |d^{4}p|((p-p_{0})¿v_{p})^{-1} j_{p}
at event p_{0} .
If the simplex is positively orientated (|d^{4}p| = d^{4}pe_{1234}^{-1})
then we have
a contribution
a d^{4}p j_{p}((p-p_{0})¿j_{p})^{-1} i^{-1} .
Rather than integrate over noninvertable 3-surface L^{-}_{p0} we might
approximate with rear tsphere O^{-}_{-h0,p0} for small scalar eventgrain h_{0} >0 , centred at p_{0} and obtain
a_{p0} = a ò_{ Oh0,p0} |d^{3}p| j_{p}((p-p_{0})¿j_{p})^{-1}
which is a sort of closeness-weighted average of j_{p}.
This gives us a "growth" equation for j_{p}
j_{p} = -(4p)^{-1} Ñ_{p}^{2}a_{p}
= -a(4p)^{-1} Ñ_{p}^{2}
ò_{ O-h0,p0} |d^{3}p| j_{p}((p-p_{0})¿j_{p})^{-1}
º H_{p}(j_{p})
where
H_{p}(y) º
ò_{ O-h0,p0} |d^{3}p| y_{p}((p-p_{0})_{*}y_{p})^{-1} i^{-1} .
We can estimate j_{p} for e^{4}¿p=t_{0} by estimating H_{p}(y) from a reasonably dense sampling of
j_{p} for e^{4}¿p<t_{0} , although we are forced to evaluate Ð_{e4}^{2} j_{p} at e^{4}¿p=t_{0}
as a "approaching" rather than a "centred" gradient estimate.
Useful zero-divergent 1-fields
Zero-gradient 1-fields
First we must consider 1-fields that are both nondivergent and solenoidal (Ñ_{p}¦(p)=0).
These trivially satisfy Ñ_{p}^{2}¦(p)=0 and are of little use for potentials since we typically require Ñ_{p}Ùa_{p} nonzero.
¦(x) = Nax + (N-2)xa = 2(N(a¿x) - xa) has Ñ_{x}¦(x)=0 for any
1-vector a.
¦(x) = (x-x_{0})|x-x_{0}|^{-N} = (x-x_{0})^{~} |x-x_{0}|^{1-N} has Ñ_{x}¦(x)=0 since Ñ_{p} (x|x|^{-k}) = (N-k)|x|^{-k} . However this undefined whenever (x-x_{0})^{2}=0. Thus for N=2 ¦(x)= |x|^{-1} x^{~} is monogenic while for N=3 we need ¦(x)= |x|^{-2} x^{~} and in N=4 ¦(x)= |x|^{-3} x^{~} .
Define a_{a,b} º a_{a,b}(x)=
a_{a,b}(x^{2}) x =
x e^{a(x2)½} (x^{2})^{½b}
= e^{ar} r^{b} x .
Then Ñ_{x} a_{a,b} =
Ñ_{x}( a_{a,b})x + N a_{a,b}
= (a a_{a,b-1}+b a_{a,b-2}) x^{2} + N a_{a,b}
= a a_{a,b+1}+(b+N) a_{a,b}
so that
Ñ_{x} a_{a,-N} = ar a_{a,-N}
Consider ¦(x)=(-x^{3})^{↑} = (-/+|x|^{3}x^{~})^{↑} according as x^{2} is ± . For x^{2}>0, |¦(x)|®0 as x®¥ . For x^{2}£0 ¦(x) = cos(|x|^{3}) + sin(|x|^{3})x^{~} is unimodular with ¦(x)=1 over L_{x}.
Also available is (l(p-p_{0})_{*}s)^{↑}a for any point-independant null 1-vector s and poinmt-independant 1-vector a orthogonal to s since Ñ_{p}(l(p-p_{0})_{*}s)^{↑}a = (l(p-p_{0})_{*}s)^{↑}(sÙa) and Ñ_{p}^{2}(l(p-p_{0})_{*}s)^{↑}a = 0 .
For an e_{4}-static 1-potential we have
r^{-2} = r^{~} |r|^{-2}
where r = ^_{e4}(p-p_{0})
well defined everywhere but along p_{0}+te_{4}.
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