For real variables p , s , a , b , c p,s,a,b,c p , s , a , b , c
K = a 2 + b 2 + c 2 − 2 ( a b + b c + c a ) , S = p 2 + s 2 , C = S + K . K=a^2+b^2+c^2-2(ab+bc+ca),\qquad S=p^2+s^2,\qquad C=S+K. K = a 2 + b 2 + c 2 − 2 ( ab + b c + c a ) , S = p 2 + s 2 , C = S + K . The shear components and wall rates are
P = K p − 2 a 2 + b 2 + c 2 + a b − 2 b c + a c , P=Kp-2a^2+b^2+c^2+ab-2bc+ac, P = K p − 2 a 2 + b 2 + c 2 + ab − 2 b c + a c , Q = K s + 3 ( − b 2 + c 2 + a b − a c ) , Q=Ks+\sqrt{3}(-b^2+c^2+ab-ac), Q = K s + 3 ( − b 2 + c 2 + ab − a c ) , r 1 = − S + 2 p , r 2 = − S − p + 3 s , r 3 = − S − p − 3 s . r_1=-S+2p,\qquad r_2=-S-p+\sqrt{3}s,\qquad r_3=-S-p-\sqrt{3}s. r 1 = − S + 2 p , r 2 = − S − p + 3 s , r 3 = − S − p − 3 s . These functions define the normalized vacuum Bianchi IX vector field
( P , Q , r 1 a , r 2 b , r 3 c ) (P,Q,r_1a,r_2b,r_3c) ( P , Q , r 1 a , r 2 b , r 3 c ) y = ( m , u , v , w ) ∈ R 4 y=(m,u,v,w)\in\mathbb R^4 y = ( m , u , v , w ) ∈ R 4
K ( u , v , w ) = u 2 + v 2 + w 2 − 2 ( u v + v w + w u ) , κ = u ( v + w − u ) , K(u,v,w)=u^2+v^2+w^2-2(uv+vw+wu),\qquad \kappa=u(v+w-u), K ( u , v , w ) = u 2 + v 2 + w 2 − 2 ( uv + v w + w u ) , κ = u ( v + w − u ) , F m = − v 2 + w 2 + u ( v − w ) − m κ , F u = 1 2 ( 1 − m 2 − ( v − w ) 2 + u 2 ) u , F_m=-v^2+w^2+u(v-w)-m\kappa,\qquad
F_u=\tfrac12(1-m^2-(v-w)^2+u^2)u, F m = − v 2 + w 2 + u ( v − w ) − mκ , F u = 2 1 ( 1 − m 2 − ( v − w ) 2 + u 2 ) u , F v = ( m − 1 − κ ) v , F w = ( − m − 1 − κ ) w , F = ( F m , F u , F v , F w ) , F_v=(m-1-\kappa)v,\qquad F_w=(-m-1-\kappa)w,\qquad
F=(F_m,F_u,F_v,F_w), F v = ( m − 1 − κ ) v , F w = ( − m − 1 − κ ) w , F = ( F m , F u , F v , F w ) , and E ( m , v ) = ( m − 1 ) 2 + v 2 E(m,v)=(m-1)^2+v^2 E ( m , v ) = ( m − 1 ) 2 + v 2
For x = ( p , s , a , b , c ) x=(p,s,a,b,c) x = ( p , s , a , b , c )
ψ ( p , z ) = 3 z 1 + p , Π ( x ) = ( ψ ( p , s ) , ψ ( p , a ) , ψ ( p , b ) , ψ ( p , c ) ) . \psi(p,z)=\frac{\sqrt3\,z}{1+p},\qquad
\Pi(x)=(\psi(p,s),\psi(p,a),\psi(p,b),\psi(p,c)). ψ ( p , z ) = 1 + p 3 z , Π ( x ) = ( ψ ( p , s ) , ψ ( p , a ) , ψ ( p , b ) , ψ ( p , c )) . For y = ( m , u , v , w ) y=(m,u,v,w) y = ( m , u , v , w ) D ( y ) = 3 + m 2 + K ( u , v , w ) D(y)=3+m^2+K(u,v,w) D ( y ) = 3 + m 2 + K ( u , v , w )
Ξ A ( y ) = ( A − 1 , A m 3 , A u 3 , A v 3 , A w 3 ) , Ξ ( y ) = Ξ 6 / D ( y ) ( y ) . \Xi_A(y)=\left(A-1,\frac{Am}{\sqrt3},\frac{Au}{\sqrt3},
\frac{Av}{\sqrt3},\frac{Aw}{\sqrt3}\right),\qquad
\Xi(y)=\Xi_{6/D(y)}(y). Ξ A ( y ) = ( A − 1 , 3 A m , 3 A u , 3 A v , 3 A w ) , Ξ ( y ) = Ξ 6/ D ( y ) ( y ) . Write N ( x ) = ( P , Q , r 1 a , r 2 b , r 3 c ) N(x)=(P,Q,r_1a,r_2b,r_3c) N ( x ) = ( P , Q , r 1 a , r 2 b , r 3 c ) C ( x ) = C ( p , s , a , b , c ) \mathcal C(x)=C(p,s,a,b,c) C ( x ) = C ( p , s , a , b , c )
The normalized and projective state spaces are R 5 \mathbb R^5 R 5 R 4 \mathbb R^4 R 4 Π , Ξ , D , F \Pi,\Xi,D,F Π , Ξ , D , F
For real k , t k,t k , t
β k ( t ) = ( m k ( t ) , 0 , v k ( t ) , 0 ) = ( 1 − k tanh ( k t ) , 0 , k cosh ( k t ) , 0 ) . \beta_k(t)=(m_k(t),0,v_k(t),0)
=\left(1-k\tanh(kt),0,\frac{k}{\cosh(kt)},0\right). β k ( t ) = ( m k ( t ) , 0 , v k ( t ) , 0 ) = ( 1 − k tanh ( k t ) , 0 , cosh ( k t ) k , 0 ) . Set
h k = arcosh k k , T k = ( k 2 + 4 ) arcosh k 3 k , d k = k 2 − 1 , h_k=\frac{\operatorname{arcosh}k}{k},\qquad
T_k=\frac{(k^2+4)\operatorname{arcosh}k}{3k},\qquad
d_k=\sqrt{k^2-1}, h k = k arcosh k , T k = 3 k ( k 2 + 4 ) arcosh k , d k = k 2 − 1 , H k ( t ) = k 2 + 4 6 t − 1 3 log cosh ( k t ) . H_k(t)=\frac{k^2+4}{6}t-\frac13\log\cosh(kt). H k ( t ) = 6 k 2 + 4 t − 3 1 log cosh ( k t ) . Use the total real conventions x / 0 = 0 x/0=0 x /0 = 0 x = max ( x , 0 ) \sqrt{x}=\sqrt{\max(x,0)} x = max ( x , 0 ) log 0 = 0 \log0=0 log 0 = 0 log x = log ∣ x ∣ \log x=\log|x| log x = log ∣ x ∣ x < 0 x<0 x < 0 arcosh x = log ( x + x 2 − 1 ) \operatorname{arcosh}x=\log(x+\sqrt{x^2-1}) arcosh x = log ( x + x 2 − 1 ) α x , 0 \alpha_{x,0} α x , 0 x x x I max ( x , 0 ) I_{\max}(x,0) I m a x ( x , 0 )
The positive transverse modes and crossing multipliers are
M k , r ( t ) = e − r t cosh ( k t ) , μ u ( k ) = e − k arcosh k , μ w ( k ) = e − 4 arcosh k / k . M_{k,r}(t)=e^{-rt}\cosh(kt),\qquad
\mu_u(k)=e^{-k\operatorname{arcosh}k},\qquad
\mu_w(k)=e^{-4\operatorname{arcosh}k/k}. M k , r ( t ) = e − r t cosh ( k t ) , μ u ( k ) = e − k arcosh k , μ w ( k ) = e − 4 arcosh k / k . Define the variational velocity with tangent forcing by
V k ( t , ξ , a ) = d F β k ( t ) ξ + a F ( β k ( t ) ) . V_k(t,\xi,a)=dF_{\beta_k(t)}\xi+aF(\beta_k(t)). V k ( t , ξ , a ) = d F β k ( t ) ξ + a F ( β k ( t )) . For ξ = ( ξ m , ξ u , ξ v , ξ w ) \xi=(\xi_m,\xi_u,\xi_v,\xi_w) ξ = ( ξ m , ξ u , ξ v , ξ w )
J k ( t , ξ ) = 2 ( m k ( t ) − 1 ) ξ m + 2 v k ( t ) ξ v − 4 v k ( t ) ξ u , J_k(t,\xi)=2(m_k(t)-1)\xi_m+2v_k(t)\xi_v-4v_k(t)\xi_u, J k ( t , ξ ) = 2 ( m k ( t ) − 1 ) ξ m + 2 v k ( t ) ξ v − 4 v k ( t ) ξ u , P k ξ = ( ξ m − ξ v / d k , ξ u , 0 , ξ w ) . P_k\xi=(\xi_m-\xi_v/d_k,\xi_u,0,\xi_w). P k ξ = ( ξ m − ξ v / d k , ξ u , 0 , ξ w ) . Write G ( y ) = 6 F ( y ) / D ( y ) G(y)=6F(y)/D(y) G ( y ) = 6 F ( y ) / D ( y )
Write X = R 5 X=\mathbb R^5 X = R 5 Y = R 4 Y=\mathbb R^4 Y = R 4 α x , t 0 \alpha_{x,t_0} α x , t 0 α ′ = N ( α ) \alpha'=N(\alpha) α ′ = N ( α ) x x x t 0 t_0 t 0 I max ( x , t 0 ) I_{\max}(x,t_0) I m a x ( x , t 0 )
J x , t 0 ( t ) = d x α x , t 0 ( t ) , Ψ t 0 ( y , t ) = Π ( α Ξ ( y ) , t 0 ( t ) ) , J_{x,t_0}(t)=d_x\alpha_{x,t_0}(t),\qquad
\Psi_{t_0}(y,t)=\Pi(\alpha_{\Xi(y),t_0}(t)), J x , t 0 ( t ) = d x α x , t 0 ( t ) , Ψ t 0 ( y , t ) = Π ( α Ξ ( y ) , t 0 ( t )) , J y , t 0 ( t ) = d Π α Ξ ( y ) , t 0 ( t ) ∘ J Ξ ( y ) , t 0 ( t ) ∘ d Ξ y . \mathcal J_{y,t_0}(t)=d\Pi_{\alpha_{\Xi(y),t_0}(t)}
\circ J_{\Xi(y),t_0}(t)\circ d\Xi_y. J y , t 0 ( t ) = d Π α Ξ ( y ) , t 0 ( t ) ∘ J Ξ ( y ) , t 0 ( t ) ∘ d Ξ y . Define x k − = Ξ ( β k ( − h k ) ) x_k^-=\Xi(\beta_k(-h_k)) x k − = Ξ ( β k ( − h k )) x k + = Ξ ( β k ( h k ) ) x_k^+=\Xi(\beta_k(h_k)) x k + = Ξ ( β k ( h k )) g ( x ) = ( Π ( x ) ) 2 g(x)=(\Pi(x))_2 g ( x ) = ( Π ( x ) ) 2 x ∗ = α x , t 0 ( t ) x_*=\alpha_{x,t_0}(t) x ∗ = α x , t 0 ( t )
ρ x , t 0 , t = d g x ∗ ( N ( x ∗ ) ) , L x , t 0 , t = − ρ x , t 0 , t − 1 d g x ∗ ∘ J x , t 0 ( t ) , \rho_{x,t_0,t}=dg_{x_*}(N(x_*)),\qquad
\mathscr L_{x,t_0,t}=-\rho_{x,t_0,t}^{-1}dg_{x_*}\circ J_{x,t_0}(t), ρ x , t 0 , t = d g x ∗ ( N ( x ∗ )) , L x , t 0 , t = − ρ x , t 0 , t − 1 d g x ∗ ∘ J x , t 0 ( t ) , H x , t 0 , t ( v ) = J x , t 0 ( t ) v + L x , t 0 , t ( v ) N ( x ∗ ) . \mathscr H_{x,t_0,t}(v)=J_{x,t_0}(t)v+
\mathscr L_{x,t_0,t}(v)N(x_*). H x , t 0 , t ( v ) = J x , t 0 ( t ) v + L x , t 0 , t ( v ) N ( x ∗ ) . Here 0 − 1 = 0 0^{-1}=0 0 − 1 = 0
Define the clock based at entry and the actual projective variation by
c k ( t ) = H k ( t ) − H k ( − h k ) , W k ( w , t ) = J β k ( − h k ) , 0 ( c k ( t ) ) w . c_k(t)=H_k(t)-H_k(-h_k),\qquad
W_k(w,t)=\mathcal J_{\beta_k(-h_k),0}(c_k(t))w. c k ( t ) = H k ( t ) − H k ( − h k ) , W k ( w , t ) = J β k ( − h k ) , 0 ( c k ( t )) w . Let Z = R 3 Z=\mathbb R^3 Z = R 3
e ( z 0 , z 1 , z 2 ) = ( z 0 , z 1 , 1 , z 2 ) , q ( m , u , v , w ) = ( m , u , w ) , χ = Ξ ∘ e , e(z_0,z_1,z_2)=(z_0,z_1,1,z_2),\qquad
q(m,u,v,w)=(m,u,w),\qquad \chi=\Xi\circ e, e ( z 0 , z 1 , z 2 ) = ( z 0 , z 1 , 1 , z 2 ) , q ( m , u , v , w ) = ( m , u , w ) , χ = Ξ ∘ e , z k − = ( 1 + d k , 0 , 0 ) , z k + = ( 1 − d k , 0 , 0 ) . z_k^-=(1+d_k,0,0),\qquad z_k^+=(1-d_k,0,0). z k − = ( 1 + d k , 0 , 0 ) , z k + = ( 1 − d k , 0 , 0 ) . For k > 1 k>1 k > 1
A k ( v 0 , v 1 , v 2 ) = ( − v 0 + 2 d k ( 1 − μ u ( k ) ) v 1 , μ u ( k ) v 1 , μ w ( k ) v 2 ) . A_k(v_0,v_1,v_2)=
\left(-v_0+\frac2{d_k}(1-\mu_u(k))v_1,\mu_u(k)v_1,\mu_w(k)v_2\right). A k ( v 0 , v 1 , v 2 ) = ( − v 0 + d k 2 ( 1 − μ u ( k )) v 1 , μ u ( k ) v 1 , μ w ( k ) v 2 ) .