For a>1 the (-lc^a)^↑ ensures convergence as t®¥ irrespective of k,l, but k<0 introduces a c^k cusp as c®0. For a=1 we require k+l<1.

    For a>0 , (-lc^-a)^↑  » 1 for large c and tends to 0 as c®0. Ghyp_k,l,-a,l accordingly diverges as t®¥ unless k+l < 0, providing damping factor (c(k+l))^↑ . k<0 aquires a c^k cusp factor at c=0 but this is swamped by the closeness of (-lc^a)^↑ to 0 for small c.

    Recall the e₄-specific hyperbolic spinor 1-vector representation with time t = x⁴ º x⁴_x º e⁴¿x ;     rapidity c º c_x,e4 º  cosh^-1(x⁴|x|^-1) ;   and 2-blade     x º x_x,e4 º xÙe₄ = XÙe₄ =Xe₄     so that for nonnull x
    x   =   |x|(-cx^~)^↑ e₄ for nonnull x and x⁴|X| = |x|²  cosh(c) sinh(c) = |x|² ½ sinh(2c) .
    Ñ_xc   =   |x|^-2(x⁴X^~ + |X|e₄)     timelike and normal to x, with
    (Ñ_xc)² = Ñ_x²c = |x|^-2     ; x(Ñ_xc) = x^~     ;   X(Ñ_xc) = |x|^-2|X|(x⁴ + x) .  
    We have used e₄ to denote the timelike direction but the following holds for Â_N-1,1 for any N³2.

    To construct e₄-specific 1-fields satisfying Ñ_x¿j_x = 0 we observe that Ñ_x(|X|^-m|x|^-nx)   =   |X|^-m|x|^-n(N-n-m - m|X|^-2x⁴XÙe₄) so
    Ñ_x¿(|X|^-m|x|^-nx)   =   0     provided n+m=N .    This motivates the e₄-dependant definition
    Q_m,x º Q_N,m,x,e4 º |X|^-m|x|^m-N       for x¹0 with
    Ñ_xQ_m,x   =   Q_m,x(-m|X|^-2X + (m-N)|x|^-2x) .
    Ñ_XQ_m,x   =   Q_m,x(-m|X|^-2 + (m-N)|x|^-2)X
    Ñ_x(Q_m,xx)   =   -m|X|^-m-3|x|^m-N x⁴x^~   =   -m|X|^-3Q_m,x x⁴x^~   =   -m|X|^-2Q_m,x x⁴x     which is 0 for m=0 and has zero scalar part for any m.
[ Proof : -m|X|^-m-2X|x|^-nx + |X|^-m(N-n)|x|^-n   =   |X|^-m|x|^-n(-m|X|^-2Xx + N-n)  .]
    Ñ_x(Q_m,xX)   =   ½Q_m,x( N-m-2 + (m-N)(-2cx^~)^↑ )     , in particular (Ñ_xQ_N-2,xX)   =   -Q_N-2,x(-2cx^~)^↑
[ Proof : (Ñ_xQ_m,x)X + Q_m,x(Ñ_xX)   =   Q_m,x(    -m|X|^-2X² + (m-N)|x|^-2xX + (N-1))   =   Q_m,x(    N-m-1 + (m-N)|x|^-2(|X|² + x⁴e₄ÙX) )
      =   Q_m,x(    N-m-1 + (m-N)( sinh(c)² -  cosh(c) sinh(c)X^~Ùe₄ )   =   Q_m,x(    N-m-1 + ½(m-N)( cosh(2c)-1  -  sinh(2c)x^~ )
      =   Q_m,x(    ½(N-m) -1 + ½(m-N)( cosh(2c) -  sinh(2c)x^~ )   =   ½Q_m,x( N-m - 2 + (m-N)(-2cx^~)^↑ )  .]

    Define j_x º r|X|^-m|x|^m-N(-lc^a)^↑ x   =   rQ_m,x(-lc^a)^↑ x     for scalar l>0 and scaling constant r>0 chosen to normalise the integration of either j⁴ =  cosh(c)r_x = rx⁴|X|^-m|x|^m-N(-lc^a)^↑ or r_x º |j_x| = r|X|^-m|x|^1+m-N(-lc^a)^↑ according to taste, over a given subspace of interest.

    Ñ_xj_x   =   (lac^a-1-m|X|^-2 coth(c) ) rQ_m,x(-lc^a)^↑ x^~   =   (-alc^a-1 + m|X|^-2 coth(c) ) (Ñ_xc)j_x   =   (-alc^a-1 + m|X|^-2 coth(c) ) (Ñ_xc)Ùj_x     has zero scalar part since Ñ_xc is orthogonal to x and so to j_x.
[ Proof : Ñ_xj_x   =   - rlac^a-1(-lc^a)^↑(Ñc)Q_m,xx + r( Ñ_xQ_m,xx)(-lc^a)^↑   =   - rQ_m,xlac^a-1(-lc^a)^↑(e₄X)^~ - r m|X|^-3Q_m,xx⁴(XÙe₄)^~ (-lc^a)^↑
      =   - rQ_m,x(-lc^a)^↑ (lac^a-1 - m|X|^-3x⁴ ) (e₄X)^~ .     (e₄X)^~ = (Ñ_xc)x .  .]
    Hence |Ñ_xj_x| = rQ_m,x (-lc^a)^↑ |lac^a-1 - m|X|^-3x⁴|
    4p|X|² r_x = 4p |X|^2-m|x|^1+m-N(-lc^a)^↑
    4p|X|² j⁴ = 4p t|X|^2-m|x|^m-N (-lc^a)^↑

Choosing The Normalisation Constant
      Holding |x|=R constant, ò _ON-1⁺-R,0 d^N-1x r_x   =   ro_N-1 R^m-N Ghyp_N-1-m,0,a,l independant of R for m=N ,
[ Proof : r ò _R^¥ dt  o_N-1|X|^N-2 |X|^-m|x|^m-N (-lc^a)^↑ o_N-1r ò _R^¥ dt |X|^N-2-m|x|^m-N (-lc^a)^↑   =   o_N-1r S^m-N ò₀^¥ dc |X|^N-1-m (-lc^a)^↑ since
  ¶c/¶t   =   ((tR^-1)²-1)^-½R^-1 = (R^-2(t²-R²))^-½R^-1 = (t²-R²)^-½ = |X|^-1     Þ dt = |X|dc  .]

      Holding t constant, ò _We4,te4 d^N-1x j⁴   =   ro_N-1 Ghyp_N-1-m,0,a,l     independant of t as expected.
[ Proof : ¶c/¶R = ((tR^-1)²-1)^-½(-tR^-2) = -t|X|^-1R^-1     Þ dR = -t^-1|X|R dc    ;     c= cosh^-1(tR^-1)    ;     R^-1 =  coshct^-1
     o_N-1r t ò ₀^t dR |X|^N-2-m|x|^m-N (-lc^a)^↑   =   o_N-1r ò_¥⁰ dc |X|^N-1-mR^1+m-N (-lc^a)^↑   =   o_N-1r ò₀^¥ dc ( sinh(c))^N-1-m (-lc^a)^↑
    Within W_e4,te4, R=|x| varies from maximum t at te₄ down to 0 as we cross L⁺₀ and then rises to ¥ outside L⁺₀. Restricting integration to within the forward nullcone corresponds to intergrating R from t down to 0.  .]

      Holding t constant, ò _We4,te4 d^N-1x r_x     =   ro_N-1 Ghyp_N-1-m,-1,a,l     independant of t.
[ Proof : o_N-1r ò ₀^t dR |X|^N-2-m|x|^1+m-N(-lc^a)^↑   =   o_N-1r t^-1 ò₀^¥ dc |X|^N-1-mR^2+m-N (-lc^a)^↑
      =   o_N-1r t^-1 ò₀^¥ dc  sinh(c)^N-1-mR^-1 (-lc^a)^↑   =   o_N-1r ò₀^¥ dc  sinh(c)^N-1-m cosh(c)^-1 (-lc^a)^↑  .]

      Holding t constant, ò _We4,te4 d^N-1x r_x  tanh(c)   =   ro_N-1 Ghyp_N-m,-2,a,l     gives the expected e₄-speed.
    j_x = (o_N-1 Ghyp_0,0,a,l)^-1 |X|^1-N |x|² (-lc^a)^↑ x^~     for m=N-1 corresponds to an ambiguous  "slow" particle; and
    j_x = (o_N-1 Ghyp_-1,0,-a,l)^-1 |X|^-N |x| (-lc^-a)^↑ x^~     for m=N corresponds to a "near light" particle having near-maximal e₄-speed.
    In both cases, Ñ_x¿j_x=0 and j_x and Ñ_xj_x vanish over L⁺₀ other than at 0 so defining j_x=0 " x : x²³0 gives j_x nonzero only inside the future nullcone L⁺₀ .

    The expected (average) four-velocity is e₄ but with a random boost of magnitude propOrtionate to (-lc^a)^↑ where positive scalar c has a Gaussian probability distribution. For large l , j_x is tightly "focussed" around e₄. Small l gives rapid dispersion.
    Throughout the above, o_k denotes the a^k-1 multiplier for the ("surface") content of a Euclidean (k-1)-sphere of radius a.

Next : General Relativity

Glossary   Contents   Author
Copyright (c) Ian C G Bell 2006
Web Source: www.iancgbell.clara.net/maths or www.bigfoot.com/~iancgbell/maths
Latest Edit: 13 Jun 2014.