In the literature, 1-curves of various types are refered to as helices. An example is

We will here use the term *helix* to refer to less general 1-curves of the form
**c** + (l**d**^{2} + w^{2}**a**^{2})^{-½}(`s`l**d** + (`s`w)^{↑}**a**) where w^{2}<0 and **a**Îw
corresponding to a linear drift **d** combined with a trigonometric circular motion in spinplane 2-blade w.
If **a**¿w ¹ 0 we have a *skew helix*.

If **e _{4}**¿w=0 then

For w^{2}>0 we have a *hyperbolic helix*.
** Timelike Helix**

Classical circular orbits are a particular perception of helical paths through spacetime.
We can regard a helical path through Â_{3,1}
as the worldine of a *longitudianally polarised* "spinless" point particle, one whose *spin* four-vector is parallel to its "averaged"
*velocity* four-vector **v**.

If **w** is a positive multiple of **v** (*ie.* **w**¿**v** < 0 for timelike **v**) the particle is said to have
*helicity* +1 and the worldine has *right-handed screw*. If **w** has opposite direction to **v**
(so that **w**¿**v** > 0 for timelike **v**)
the particle has helicity -1 and the worldine has *left handed screw*.

Neutrinos are considered left-handed (*ie.* negative helicity) with right-handed *antineutrinos* far rarer.
Left-handed fermions appear more efficient at mediating the *weak interactions*, suggesting
a violation of *chiral* or *mirror* symmetry of our (local) universe.

Let *F*_{0} = { c**e _{4}** : c Î Â } be the worldline of a "Fixed" observer.
Let

**p**_{F}(c) =
*S**e*^{wce21}**e _{1}** + c

Rather than twice the radius, the *diameter* of a helix will here refer to the seperation between
two events on the helix **e _{4}**-temporally seperated by half the period, such as the seperation

(*d***p**)^{2} =
(*S*w*d*c)^{2} - *d*c^{2}
= ((*S*w)^{2} - 1)*d*c^{2} so the proper time
formulation of ** P** is given by t = t

g = (1-

Setting unit spinor R

h =

The *proper period* is thus (2p)(gw)^{-1} with an associated net timelike drift
of (2p)w^{-1} . The acceleration **p**"(t) is spacelike, being
(gw)^{2}*S* = `H`^{2}*S*^{-3} times the instantaneous inward unit radial vector,
where angular momentum `H`=g*S*^{2}w.

For large g, w » *S*^{-1} , `H`» g*S* and the acceleration approaches g*S*^{-2}.

Because **e _{4}**

This works because the "drift"

For a more general "skewed helix" with nonunit drift **d** containing a component within unit
**b**; passing through **a** with instantaneous spin centre **0**
( so **a**^{2}=*S*^{2}) we have factored spinor form

** p**(c) = (
(½c

(**a**+**d**)^{↑}_{§}
= ((½*e*_{¥}
(¼wc**b**)^{↑}_{§}(c¯_{b}(**d**))
)^{↑}**a**^{↑})_{§}
= (
(¼wc**b**)^{↑}_{§}(**d**
)^{↑}**a**^{↑})_{§}
so that (**a**+**d**)^{↑}_{§} is equivalent to the helix **a**^{↑}_{§} with an additional
displacement by c¯_{b}(**d**) rotated in **b** at half the helical turn rate. This additional displacement is
an outward spiral with period c_{T} = 4pw^{-1} and radius c|¯_{b}(**d**)| .

[ Proof :
**d** anticommutes with **a** and **b**
but commutes with <0;4>-vector
**a**^{2} = ¼c^{2}w(w**b**^{2} - 2*e*_{¥}^_{b}(**d**)**b**)
while **a** trivially commutes with **d**^{2}=0 .
Set **d'**=¯_{b}(**d**)**b**, *ie.* ¯_{b}(**d**) rotated by ½p in **b**.
Null 2-blade **d'**= *e*_{¥}**d'** anticommutes with **d** and **a**.

Null 4-blade A=w*e*_{¥}^_{b}(**d**)**b** commutes with **a** and **d**
and **a**^{2}=(½cw)^{2}(**b**^{2}-2A) suggests expressing
**a**=(½cw)(**b**^{2}-2A)^{½} a' where
a'=(½cw)^{-1} (**b**^{2}-2A)^{-½} **a**
has a'^{2}=1. If **b**^{2}=1 we have
(**b**^{2}-2A)^{-½} = 1+A but for spacial **b** we require an i commuting with **a** and **d**
to form (**b**^{2}-2A)^{-½} = i(1-A).
If ^_{b}(**d**)^{2}=1 then 5-blade i=(e_{¥0}Ù**d**Ù**b**)^{~} suffices while for
timelike ^_{b}(**d**) we can resort to the 7-blade pseudoscalar of Â_{4,1}^{%} .

Setting c = (½cw)^{-1} (**b**^{2}-2A)^{-½} **d**
we have **a**+**d** =
(½cw)(**b**^{2}-2A)^{½} (a'+c) where a'^{2}=1 and c^{2}=0,
with a' and c anticommuting so that
(a'+c)^{↑} = a'^{↑} + c *sinh*1
= *cosh*(1) + *sinh*(1)(a' + c')

Clearly
the 2-blade **da**^{-1}
= -(½cw**b**^{2})^{-1} w**db**
= -**b**^{-2}w^{-1}**d'**
anticommutes with **a** and 2-vector *sinh*(**a**) = (**a**^{↑})_{<2>}
but commutes with **a**^{2} and *cosh*(**a**) and we have
**d'****a**^{↑} = **a**^{-↑}**d'**.
Thus because **a** and **d** anticommute and **d**^{2}=0 ; and
(**a**±**d**)^{2k} = **a**^{2k} and
(**a**±**d**)^{2k+1} = (**a**±**d**)**a**^{2k}
so (**a**±**d**)^{↑} = **a**^{↑} ±
**d**(1+3!^{-1}**a**^{2} + 5!^{-1}**a**^{4} + ....)
= **a**^{↑} ± **da**^{-1} *sinh*(**a**) ;
we have

(**a**±**d**)^{↑}
= **a**^{↑} ± **da**^{-1} *sinh*(**a**)
= **a**^{↑} -/+ *sinh*(**a**) **da**^{-1}
and hence

(**a**+**d**)^{↑} x (-**a**-**d**)^{↑}
= (**a**^{↑} + **da**^{-1} *sinh*(**a**))x(**a**^{-↑} - **da**^{-1} *sinh*(**a**))
= (**a**^{↑} + **da**^{-1} *sinh*(**a**))x(**a**^{-↑} + *sinh*(**a**)**da**^{-1})

=
**a**^{↑}x**a**^{-↑}
+ **ba**^{-1} *sinh*(**a**)x**a**^{-↑}
+ **a**^{↑}x *sinh*(**a**)**ba**^{-1}
+ **ba**^{-1} *sinh*(**a**)x *sinh*(**a**)**ba**^{-1}

=
( **a**^{↑}x**a**^{-↑}
+ ½**da**^{-1} **a**^{↑}x**a**^{-↑}
- ½ **a**^{↑}x**a**^{-↑} **da**^{-1}
- ¼ **da**^{-1} **a**^{↑}x**a**^{-↑} **da**^{-1} )
+ (½ **a**^{↑}x**a**^{↑} **da**^{-1} + ¼ **da**^{-1} **a**^{↑}x**a**^{↑} **da**^{-1} )
+ (-½ **da**^{-1} **a**^{-↑}x**a**^{-↑} + ¼ **da**^{-1} **a**^{-↑}x**a**^{-↑} **da**^{-1} )

=?=
((¼wc**b**)^{↑}_{§}(**d**))^{↑}
**a**^{↑}x**a**^{-↑}
((¼wc**b**)^{↑}_{§}(**d**))^{-↑}

= (½*e*_{¥}(¼wc**b**)^{↑}_{§}(c¯_{b}(**d**))^{↑}
**a**^{↑}x**a**^{-↑}
(½*e*_{¥}(¼wc**b**)^{↑}_{§}(c¯_{b}(**d**))^{-↑}

Also
**a**^{↑}**d**^{↑}
= *cosh*(**d**) a^{↑} + *sinh*(**d**) (-a)^{↑}
= a^{↑} + **d**(-a)^{↑}
**d**^{↑}**a**^{↑}
= a^{↑} *cosh*(**d**) + (-a)^{↑} *sinh*(**d**)
= a^{↑} + (-a)^{↑} **d**
since **d**^{2}=0 .
.]

Each orbit appears to ** P** to take

At

Now consider two **F**-fixed marker objects at *S***e _{1}** and

More generally, suppose (**f _{1}**,

In compiling a history of

The *null helix* has w = ±*S* , g=¥.

**p**(`t`) = (½w`t`e_{12})^{↑}_{§}(*S***e _{1}**+

Even though g=¥ for a null helix, we can frequently obtain expressions for the null helix by setting w=

The

A null helix has no proper time parameterisation; the

The *spacelike helix* with |*S*w| > 1 has
**p**(t) = R_{t}_{§}(*S***e _{1}** + tg

An external **e _{4}**-observer percieves a particle traversing a 2p

A spacelike helix having *S*w = ±½p ; g=(¼p^{2}-1)^{-½}
has zero diamter.

The Ashworth model of a photon of wavelength l is a chargeless particle of nonzero rest mass `m` traversing a spacelike helix
of radius (2p)^{-1}l and null drift. The speed of the photon is 2^{½}`c` making its kinetic energy ½`m``v`^{2} =
`m``c`^{2} which we can equate with *h*`c`l^{-1} if we set `m`=`m`(l)=*h*(`c`l)^{-1}
** Spacelike Helix - Spacelike Drift**

**p**(c) = c**e _{3}** +

A null helix has acceleration w^{2}*S* equal to frequency w .

Setting unit spinor R_{t} *º* R(t) º (½wt b)^{↑}
we have

**p**(t) = R_{t}_{§}(*S***a** + t**s**)
= *S*R_{t}_{§}(a) + t**s**
where null **s** is orthonal to unit spacelike **a**.

**p**'(t)
= R_{t}_{§}(**a** + **s**)
with **p**'(t)^{2} = **a**^{2}.

**p**"(t) = -wR_{t}_{§}(**a**)
with **p**"(t)^{2} = w^{2} .

**p**"(t)**p**(t) = -w R_{t}_{§}(*S*we_{12} + e_{14})
= -wb - wR_{t}_{§}(**a**Ù**s**)
= -gh^{2}we_{12} - g^{2}hw R_{t}_{§}(e_{14})
where

h = *S*wg = ((*S*w)^{-2}-1)^{-½} = g(1-g^{2})^{½} .

" **p**'(t)^{2} = R_{t}_{§}(**a**^{2}+**s**^{2}+2)

The *proper period* is thus (2p)w^{-1} with an associated null drift
of (2p)w^{-1} . The acceleration **p**"(t) is spacelike, being
w times the instantaneous inward unit radial vector,

If **a**¿**s**¹0 we have **p**'(t)^{2}=**a**^{2}+2**a**¿**s** allowing a natural parameterisation

For a skew helix ... ?

Next : Spacetime Kinematics

Glossary Contents Author

Copyright (c) Ian C G Bell 1999, 2014

Web Source: www.iancgbell.clara.net/maths

Latest Edit: 15 Jun 2014.