We will here use the term helix to refer to less general 1-curves of the form c + (ld² + w²a²)^-½(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)^-½ (le₃   + (swe₁(2^½e₂+e₄))^↑e₁) having drift (l²+w²)^-½le₃   , spinplane e₁(2^½e₂+e₄) , radius (l²+w²)^-½ and spacial period 2pw^-1. We have p(s)=c + s(l²+w²)^-½le₃ + (l²+w²)^-½ cos(sw)e₁ + (l²+w²)^-½ sin(sw))(2^½e₂+e₄) so the e₁ and e₂ coordinates trace an origin centred ellipse through e₁ and 2^-½e₂  while e⁴¿p(s) oscillattes within ± (l²+w²)^-½ 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)^-½ 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 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 F₀ = { ce₄ : c Î Â } be the worldline of a "Fixed" observer. Let P = { p_F(c) : c Î Â } be the worldline of an "orbitting" observer where

      p_F(c)   =   Se^wce₂₁e₁ + ce₄   =   S(wce₂₁)^↑e₁ + ce₄   =   (½wc e₂₁)^↑_§(Se₁ + ce₄)   =   e^½wce₂₁(Se₁ + ce₄)e^-½wce₂₁   =   (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 T_F = 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 p_F(0)-p_F(-w^-1p)   =   Se₁ - (-Se₁±p(wg_w^-1)g_we₄)   =   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 = t_P = gc where positive scalar relativity factor  
    g = (1-S²w²)^-½ » 1 + ½S²w² + (3/8)S⁴w⁴ + _O((Sw)⁶).
    P = { SR_P(t)_§(e₁) + tge₄ : t_P Î Â } = { R_P(t)_§(Se₁ + tge₄) : t_P Î Â }
    Setting unit spinor R_t º R(t) º (½wgt e₂₁)^↑ we have
       p(t)   =   R_t§(Se₁ + tge₄)   =   SR_t§(e₁) + tge₄ ;
    p'(t)   =   gR_t§(Swe₂ + e₄)   =   gSwR_t§(e₂) + ge₄     [ p'(t)²=-1 ] ;
    p"(t)   =    -g²w²S R_t§(e₁)        with p"(t)² = g⁴w⁴S² ;
      p"(t)p(t)   =   -g³w²S R_t§(Swe₁₂ + e₁₄)   =   -g³w³S²e₁₂ - g³w²S R_t§(e₁₄)   =   -gh²we₁₂ - g²hw R_t§(e₁₄)     where
    h = Swg   = ((Sw)^-2-1)^-½ = 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+e₀+½S²e_¥)    .
    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))^↑ (½c(e_¥^_b(d)-wb)^↑ )_§(a)   =   (d^↑a^↑)_§(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_¥ (¼wcb)^↑_§(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 c_T = 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)^-½ a has a'²=1. If b²=1 we have (b²-2A)^-½ = 1+A but for spacial b we require an i commuting with a and d to form (b²-2A)^-½ = 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 Â_4,1^% .
    Setting c = (½cw)^-1 (b²-2A)^-½ 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 = a^2k and (a±d)^2k+1 = (a±d)a^2k 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)
      =   a^↑xa^-↑ + ba^-1 sinh(a)xa^-↑ + a^↑x sinh(a)ba^-1 + ba^-1 sinh(a)x sinh(a)ba^-1
      =   ( a^↑xa^-↑ + ½da^-1 a^↑xa^-↑ -   ½ a^↑xa^-↑ da^-1 -   ¼ da^-1 a^↑xa^-↑ da^-1 ) + (½ a^↑xa^↑ da^-1 + ¼ da^-1 a^↑xa^↑ da^-1 ) + (-½ da^-1 a^-↑xa^-↑  + ¼ da^-1 a^-↑xa^-↑ da^-1 )
    =?= ((¼wcb)^↑_§(d))^↑ a^↑xa^-↑ ((¼wcb)^↑_§(d))^-↑
      =   (½e_¥(¼wcb)^↑_§(c¯_b(d))^↑ a^↑xa^-↑ (½e_¥(¼wcb)^↑_§(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 d²=0 .  .]

    Each orbit appears to P to take P-time (t interval) T_P = 2p(gw)^-1 which is less than T_F = T_P(1-S²w²)^-½ = 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+t_Fe₄, 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 p_E(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 dg_V^-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 F_p so that f_i=F_pe_iF_p^§.
    In compiling a history of Q, observer F projects spacially into _f123 = F_p§(e₁₂₃) observes a 1-conic ¯_Rp§(e123)(SR_t§(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|^-1(½wte₁₂)^↑_§(e₁) +te₄ with null velocity p'(t) = (½wte₁₂)^↑_§(e₂+e₄) ; spacelike acceleration p"(t) = -w(½w^-1te₁₂)^↑_§(e₁) ; and null p"(t)p'(t) = -w(½wte₁₂)^↑_§(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+e₀+½S²e_¥)    .
    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)   =   R_t§(Se₁ + tge₄)   =   SR_t§(e₁) + tge₄ where g = ((Sw)²-1)^-½ now has the range (0,¥) .
    p'(t)   =   gR_t§(Swe₂ + e₄)   =   gSwR_t§(e₂) + ge₄     with p'(t)²=1 ;
    p"(t)   =    -g²w²S R_t§(e₁)        with p"(t)² = (gw)⁴S² ;
      p"(t)p(t)   =   -g³w²S R_t§(Swe₁₂ + e₁₄)   =   -g³w³S²e₁₂ - g³w²S R_t§(e₁₄)   =   -gh²we₁₂ - g²hw R_t§(e₁₄)     where h = Swg   = ((Sw)^-2-1)^-½ .

    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)^-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)^-1l and null drift. The speed of the photon is 2^½c 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 Lift_e4^-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 R_t º R(t) º (½wt b)^↑ we have
       p(t)   =   R_t§(Sa + ts)   =   SR_t§(a) + ts where null s is orthonal to unit spacelike a.
    p'(t)   =   R_t§(a + s) with p'(t)² = a².
    p"(t)   =    -wR_t§(a)        with p"(t)² = w² .
      p"(t)p(t)   =   -w R_t§(Swe₁₂ + e₁₄)   =   -wb - wR_t§(aÙs)   =   -gh²we₁₂ - g²hw R_t§(e₁₄)     where
    h = Swg   = ((Sw)^-2-1)^-½ = g(1-g²)^½ .

" p'(t)² = R_t§(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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Web Source: www.iancgbell.clara.net/maths
Latest Edit: 15 Jun 2014.