Helices
In the literature, 1-curves of various types are refered to as helices. An example is p(s)=c + (s2+w2)( sw-1(swe12)e1 + ws-1(sse34)e3) . This spacelike curve has p'(s)2=1 and e4¿p(s) monontonically increasing from -¥ to +¥ with s while e3¿p(s) is minimised at s=0 by c3 + (s2+w2) ws-1 . Meanwhile the e1 and e2 coordinates circle about ¯e12(c) with radius sw-1 and length period 2pw-1.   We can think of this as a circular motion with a forward hyperbolic drift.

We will here use the term helix to refer to less general 1-curves of the form c + (ld2 + w2a2)(sld + (sw)a) where w2<0 and aÎw 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 e4¿w=0 then e4¿p(s) = sd4 and as we add e4 into w we obtain a temporal oscillation which may exceed the d4 drift and render e4¿p(s) nonmontonic. As an example consider p(s)=(l2+w2) (le3   + (swe1(2½e2+e4))e1) having drift (l2+w2)le3   , spinplane e1(2½e2+e4) , radius (l2+w2) and spacial period 2pw-1. We have p(s)=c + s(l2+w2)le3 + (l2+w2) cos(sw)e1 + (l2+w2) sin(sw))(2½e2+e4) so the e1 and e2 coordinates trace an origin centred ellipse through e1 and 2e2  while e4¿p(s) oscillattes within ± (l2+w2) with the same period (2p)-1w as the e12 ellipse. Observers will measure the extent of this "temporal wobble" in units c-1 (l2+w2) with the same period (2p)-1w as the e12 ellipse.

For w2>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 Â3,1 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 F0 = { ce4 : c Î Â } be the worldline of a "Fixed" observer. Let P = { pF(c) : c Î Â } be the worldline of an "orbitting" observer where

pF(c)   =   Sewce21e1 + ce4   =   S(wce21)e1 + ce4   =   (½wc e21)§(Se1 + ce4)   =   e½wce21(Se1 + ce4)ewce21   =   (½wce21)(Se1 + ce4)(-½wce21)   =   S( cos(wc)e1 + sin(wc)e2) + ce4     with (Sw)2<1 is the timelike 1-curve worldline of a particle circling F at distance S > 0 in the e12=e1Ùe2 spatial plane in "counterclockwise" direction (rotating e1 into e2 as c increases). It is percieved by F to complete one orbit of radius S in time interval TF = 2pw-1.

Rather than twice the radius, the diameter of a helix will here refer to the seperation between two events on the helix e4-temporally seperated by half the period, such as the seperation pF(0)-pF(-w-1p)   =   Se1 - (-Se1±p(wgw-1)gwe4)   =   2Se1 ± pw-1e4 having square 4S2-p2w-2 £ S2(4-p2) for timelike helix with equality for the null helix with Sw=±1.

(dp)2 = (Swdc)2 - dc2 = ((Sw)2 - 1)dc2 so the proper time formulation of P is given by t = tP = gc where positive scalar relativity factor
g = (1-S2w2) » 1 + ½S2w2 + (3/8)S4w4 + O((Sw)6).
P = { SRP(t)§(e1) + tge4 : tP Î Â } = { RP(t)§(Se1 + tge4) : tP Î Â }
Setting unit spinor Rt º R(t) ºwgt e21) we have
p(t)   =   Rt§(Se1 + tge4)   =   SRt§(e1) + tge4 ;
p'(t)   =   gRt§(Swe2 + e4)   =   gSwRt§(e2) + ge4     [ p'(t)2=-1 ] ;
p"(t)   =    -g2w2S Rt§(e1)        with p"(t)2 = g4w4S2 ;
p"(t)p(t)   =   -g3w2S Rt§(Swe12 + e14)
=   -g3w3S2e12 - g3w2S Rt§(e14)   =   -gh2we12 - g2hw Rt§(e14)     where
h = Swg   = ((Sw)-2-1) = g(1-g2)½ .

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)2S = H2S-3 times the instantaneous inward unit radial vector, where angular momentum H=gS2w.
For large g, w » S-1 , H» gS and the acceleration approaches gS-2.

Because e4e¥ = e4Ùe¥ commutes with e12 we can "fully spinorise" this form in GHMST (see below) as
p(t)   =   (½gt(we21+e¥4))§(Se1+e0S2e¥)    .
This works because the "drift" e4 is perpendicular to the "spin plane" e12 but even so the square (-½gt(we21+e¥4))2 = ¼g2t2( -w2-2we¥124) has a 4-blade component.

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 a2=S2) we have factored spinor form
p(c) = ( (½ce¥d) (-½cwb) )§(a)   =   ( (½ce¥¯b(d))c(e¥^b(d)-wb) )§(a)   =   (da)§(a)     where c is an "external" rather than "proper" time parameter; 2-vector a = a(c) = ½c(e¥^b(d) - wb) has a-1 = -(½cw2b2)-1 (e¥^b(d)+wb) = (½cwb)-2 a(#) ; and null 2-blade d = d(c) = ½ce¥¯b(d) .

(a+d)§   =   ((½e¥wcb)§(c¯b(d)) )a)§   =   ( (¼wcb)§(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 cT = 4pw-1 and radius c|¯b(d)| .
[ Proof : d anticommutes with a and b but commutes with <0;4>-vector a2 =   ¼c2w(wb2 - 2e¥^b(d)b) while a trivially commutes  with d2=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=we¥^b(d)b commutes with a and d and a2=(½cw)2(b2-2A) suggests expressing a=(½cw)(b2-2A)½ a' where a'=(½cw)-1 (b2-2A) a has a'2=1. If b2=1 we have (b2-2A) = 1+A but for spacial b we require an i commuting with a and d to form (b2-2A) = i(1-A). If ^b(d)2=1 then 5-blade i=(e¥0ÙdÙb)~ suffices while for timelike ^b(d) we can resort to the 7-blade pseudoscalar of Â4,1% .
Setting c = (½cw)-1 (b2-2A) d we have a+d = (½cw)(b2-2A)½ (a'+c) where a'2=1 and c2=0, with a' and c anticommuting so that (a'+c) = a' + c sinh1 =  cosh(1) +  sinh(1)(a' + c')
Clearly the 2-blade da-1   =   -(½cwb2)-1 wdb   =   -b-2w-1d' anticommutes with a and 2-vector  sinh(a) = (a)<2> but commutes with a2 and  cosh(a) and we have d'a = a-↑d'. Thus because a and d anticommute and d2=0 ; and (a±d)2k = a2k and (a±d)2k+1 = (a±d)a2k so (a±d)   =   a ± d(1+3!-1a2 + 5!-1a4 + ....)   =   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)
=   axa-↑ + ba-1 sinh(a)xa-↑ + ax sinh(a)ba-1 + ba-1 sinh(a)x sinh(a)ba-1
=   ( axa-↑ + ½da-1 axa-↑ -   ½ axa-↑ da-1 -   ¼ da-1 axa-↑ da-1 ) + (½ axa da-1 + ¼ da-1 axa da-1 ) + (-½ da-1 a-↑xa-↑  + ¼ da-1 a-↑xa-↑ da-1 )
=?= ((¼wcb)§(d)) axa-↑ ((¼wcb)§(d))-↑
=   (½e¥wcb)§(c¯b(d)) axa-↑e¥wcb)§(c¯b(d))-↑

Also ad     =    cosh(d) a +  sinh(d) (-a)   =   a + d(-a) da     =   a  cosh(d) + (-a)  sinh(d)   =   a + (-a) d since d2=0 .  .]

Each orbit appears to P to take P-time (t interval) TP = 2p(gw)-1 which is less than TF = TP(1-S2w2) = 2p w-1 .
At E-time 0, E has spacetime position Se1 and is percieved by F0 to have velocity v = Swe2. E percieves F to have position   ^(0+tFe4, e4)=0 so E percieves F as a maintaining constant distance S from him. Even though it does appear to E that F0 is orbiting E, this is not a symmetrical situation since d2 pE(t) / dt2 is nonzero whereas F's worldline is inertial (ie. a straight path).

Now consider two F-fixed marker objects at Se1 and Se1 + de2. F percieves these as having spatial seperation d but at E-time 0 E percieves their spacial seperation as dgV-1. Postulating such marker objects around the entire orbit and letting d ® 0 we see that E considers the orbit to have radius S but circumference 2pS(1-S2w2)½ < 2pS . In the general relativistic paradigm, we say space appears to accelerating observer E to be "warped".
E and F agree on the orbital "radius" S and "speed" Sw but disagree on the length and period of the orbit.

More generally, suppose (f1,f2,f3,f4) is a frame for an observer F at event 0 with f4=(e4+V)~ for spacelike V Î e4* with V2<1 . Define rotor Fp so that fi=FpeiFp§.
In compiling a history of Q, observer F projects spacially into _f123 = Fp§(e123) observes a 1-conic ¯Rp§(e123)(SRt§(e1)) combined with a drift velocity tg¯Rp§(e123)(e4) parallel to the principle axis.

Null Helix

The null helix has w = ±S , g=¥.
p(t)   =   (½wte12)§(Se1+te4)   =   |w|-1wte12)§(e1) +te4 with null velocity p'(t) = (½wte12)§(e2+e4) ; spacelike acceleration p"(t) = -ww-1te12)§(e1) ; and null p"(t)p'(t) = -wwte12)§(e12+e14) .
Even though g=¥ for a null helix, we can frequently obtain expressions for the null helix by setting w=S-1 and g=1 in formula for the timelike helix.
The GHMST form is p(t)   =   (½t(±S-1e21+e¥e4))§(Se1+e0S2e¥)    .
A null helix has no proper time parameterisation; the t in the above is helical axis (e4) time. A fixed e4 observer considers the orbital period to be 2pS and the circumference to be 2pS and so considers the spacial speed of the particle to be unity.

Spacelike Helix - Timelike Drift

The spacelike helix with |Sw| > 1 has p(t)   =   Rt§(Se1 + tge4)   =   SRt§(e1) + tge4 where g = ((Sw)2-1) now has the range (0,¥) .
p'(t)   =   gRt§(Swe2 + e4)   =   gSwRt§(e2) + ge4     with p'(t)2=1 ;
p"(t)   =    -g2w2S Rt§(e1)        with p"(t)2 = (gw)4S2 ;
p"(t)p(t)   =   -g3w2S Rt§(Swe12 + e14)   =   -g3w3S2e12 - g3w2S Rt§(e14)   =   -gh2we12 - g2hw Rt§(e14)     where h = Swg   = ((Sw)-2-1) .

An external e4-observer percieves a particle traversing a 2pS circumference in time (2p)-1 w-1 and deduces speed |Sw| > 1 .

A spacelike helix having Sw = ±½p ; g=(¼p2-1) 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)-1l and null drift. The speed of the photon is 2½c making its kinetic energy ½mv2 = mc2 which we can equate with hcl-1 if we set m=m(l)=h(cl)-1

Spacelike Helix - Spacelike Drift

p(c) = ce3 + S( cos(wc)e1+ sin(wc)e4) satisfying (p(c)-ce3)(p(c)-ce3) = (p(c)-ce3)¿+(p(c)-ce3) = S2 where is e4-dependant Hermitian conjugation is formally definable as Lifte4-1(((½wc e41')§(Se1' + ce3')) where (e1',e2',e3',e4') is a basis for Â4, but being e4-dependant is of limited use.

Spacelike Helix - Null Drift

A null helix has acceleration w2S equal to frequency w .
Setting unit spinor Rt º R(t) ºwt b) we have
p(t)   =   Rt§(Sa + ts)   =   SRt§(a) + ts where null s is orthonal to unit spacelike a.
p'(t)   =   Rt§(a + s) with p'(t)2 = a2.
p"(t)   =    -wRt§(a)        with p"(t)2 = w2 .
p"(t)p(t)   =   -w Rt§(Swe12 + e14)
=   -wb - wRt§(aÙs)   =   -gh2we12 - g2hw Rt§(e14)     where
h = Swg   = ((Sw)-2-1) = g(1-g2)½ .

" p'(t)2 = Rt§(a2+s2+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¿s¹0 we have p'(t)2=a2+2a¿s allowing a natural parameterisation

For a skew helix ... ?

Next : Spacetime Kinematics

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Latest Edit: 15 Jun 2014.