Michael Taylor defines timelike in his book Pseudodifferential Operators (1981, Princeton Legacy Library, Chapter 4, §4, Finite propagation speed) as follows: Let $L$ be a strictly hyperbolic differential operator. That means for $$L:=\partial_t^m-\sum_{j=0}^{m-1}A_j(t,x,D_x) \partial_t^{j}$$ and its principle symbol $$L_m(t,x,\tau,\xi):=(i\tau)^m -\sum_{j=1}^{m-1} \widetilde A_{m-j}(t,x,\xi) (i\tau)^j,$$ all roots of $L_m$ are real and distinct. Then a vector $V:=(V_0,V')$ at $(t_0,x_0)$ is timelike with respect to $L$ if the equation $$L_m(t_0,x_0,X+\tau V)=0,$$ has distinct real roots again for all $X$ not proportional to $V$. So far so good. Now the question. On the next page the following is claimed: " Indeed, if $V=(V_0,V')$ and if $$\frac{|V_0|}{|V'|}>\frac{|\tau|}{|\xi|}$$ with $(\tau,\xi)$ with $L_m(t_0,x_0,\tau,\xi)=0,\xi\neq0$ then $V$ is timelike."
I cant see the connetion here. Can anyone help me please? :-)