How can I draw a sphere in the Minkowski space?

If the equation of the circle in the Minkowski 3 space is given as
$S_1^2 = {x in E_1^3:- x_1^2 + x_2^2 + x_3^2 }$, how can I replace it in the following code?

f[u_] :=
 (eqns = {t'[s] == κ[s] n[s], 
    n'[s] == -κ[s] t[s] + τ[s] b[s], 
    b'[s] == -τ[s] n[s], r'[s] == t[s], t[0] == t0, n[0] == n0, 
    b[0] == b0, r[0] == r0};
  Subscript[v, x0] = u;
  α = .4; β = 0.5;
  κ[s_] = 2 β Sech[u α + s β];
  τ[s_] = 1 - α/β;
  {t0, n0, b0} = Orthogonalize[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}];
  r0 = {0, 0, 0};
  sol = First@NDSolve[eqns, {r, t, n, b}, {s, 0, 20}];
  ParametricPlot3D[Evaluate[r[s] /. sol], {s, 0, 20}, 
   PlotStyle -> {Thick, Red}, PlotRangePadding -> 1, Boxed -> False, 
   Axes -> None
   ])

gm[u_] :=
 (eqns = {t'[s] == κ[s] n[s], 
    n'[s] == -κ[s] t[s] + τ[s] b[s], 
    b'[s] == -τ[s] n[s], r'[s] == t[s], t[0] == t0, n[0] == n0, 
    b[0] == b0, r[0] == r0};
  α = .4; β = 0.5;
  κ[s_] = 2 β Sech[u α + s β];
  τ[s_] = 1 - α/β;
  {t0, n0, b0} = Orthogonalize[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}];
  r0 = {0, 0, 0};
  sol = First@NDSolve[eqns, {r, t, n, b}, {s, 0, 20}];
  Graphics3D[
   Text[Style["t=" <> ToString[u], Black, Italic, 
     30], {Evaluate[r[20] /. sol]}]])
Show[Table[f[u], {u, 0, 4}], Table[gm[u], {u, 0, 4}], PlotRange -> All]

g1[u_] := (eqns = {t'[s] == κ[s] n[s], 
     n'[s] == -κ[s] t[s] + τ[s] b[s], 
     b'[s] == -τ[s] n[s], r'[s] == t[s], t[0] == t0, n[0] == n0, 
     b[0] == b0, r[0] == r0};
   α = .4; β = 0.5;
   κ[s_] = 2 β Sech[u α + s β];
   τ[s_] = 1 - α/β;
   {t0, n0, b0} = Orthogonalize[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}];
   r0 = {0, 0, 0};
   sol = First@NDSolve[eqns, {r, t, n, b}, {s, 0, 20}];
   ParametricPlot3D[Evaluate[t[s] /. sol], {s, 0, 20}, 
    PlotStyle -> {Thick, Blue}, PlotRange -> All, Boxed -> False, 
    Axes -> None]);
sph = SphericalPlot3D[{1}, {θ, 0, Pi}, {ϕ, 0, 2 Pi}, 
   Boxed -> False, Axes -> None, 
   PlotStyle -> 
    Directive[Orange, Opacity[0.7], Specularity[White, 10]], 
   Mesh -> None, PlotPoints -> 10];

where in the above code, we draw an evolved curve on a sphere in the Euclidean space. In the Minkowski space (timelike type) we have Frenet formulae as:

$qquad t’ = kappa,n,quad n’ = kappa,t + tau,b, quad b’ = – tau,n$.

Also, $<t,,t>, = -1$, $<n,,n>, = 1$, $<b,b>, = 1$, $t times n = b$, $n times b = – t$ and $b times t = n$. The inner product is defined as

$qquad <x,y>, = – x_{1}, y_{1} + x_{2}, y_{2} + x_{3}, y_{3}$

where

$qquad x =, <x_{1},x_{2},x_{3}> quad y =, <y_{1} ,y_{2},y_{3}>$.

and the cross product is

$qquad x , times , y = begin{vmatrix} -e_{1} & e_{2} & e_{3} x_1 & x_2 & x_3 y_1 & y_2 & y_3 end{vmatrix}$.

Can any one help?. I need applying the same code above in the Minkowski space.

ContourPlot3D[-x^2 + y^2 + z^2 == 1, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]