$\gamma$ is a smooth map $$A=(A_0,\dots A_{n+1}):I\to \mathrm{A}^{\uparrow}_+(2,n)$$ such that $A_0$ is a null lift and $A'_0\in \mathrm{Span}(A_0,A_1)$. First-order frame fields do exist along any time-like curve. If $A$ is a first-order frame, then any other is given by $$\widetilde{A}=A\cdot X(r,x,y,R),$$ where $$r,x : I\to \R, x>0,\quad y:I\to \R^{n-1},\quad R:I\to \mathrm{SO}(n-1)$$ are smooth functions and $$\label{gauge}X(r,x,y,R)=\left(
               \begin{array}{cccc}
                 r &-x & ^ty\cdot R & \frac{^ty\cdot y-x^2}{2} \\
                 0 & 1 & 0 & x/r \\
                 0 & 0 & R & y/r \\
                 0 & 0 & 0 & r^{-1} \\
               \end{array}
             \right).$$

[A first-order frame is said of the [*second-order*]{} if $$A_0=M_0,\quad A_1=M_1,\quad A_2=M_2,\quad A_{n+1}=M_{n+1}.$$ Second-order frames do exist along any generic, time-like curve with its intrinsic orientation. Note that $(A_3,\dots A_n)$ is a trivialization of the third-order normal bundle of $\gamma$.]{}

If $A$ is a second order frame field, then $$\label{LS}A' = A\cdot \left(
              \begin{array}{ccccc}
                0 & -h_1& 1 & 0 & 0 \\
                1 & 0 & 0 & 0 & h_1 \\
                0 & 0 & 0 & -^ts & 1 \\
                0 & 0 & s & \phi & 0 \\
                0 & -1 & 0 & 0 & 0 \\
              \end{array}
            \right)\upsilon_{\gamma},$$ where $\phi =(\phi^j_i)_{i,j=3,\dots,n} 