First some nomenclature:
Let $Q(s,t): [0,1]\times[0,1]$ $\rightarrow$ $L^4$ be a smooth map. Let $x^i$ be affine coordinates in $L^4$. For each $t$ we define a longitudinal curve of $Q$ by $\gamma_t(s) = Q(s,t)$. For each $s$ we define a transverse curve of $Q$ by $\gamma_s(t) = Q(s,t)$. The map $V:[0,1] \rightarrow TL^4$ which assigns to each $s$ the velocity of the transverse curve at $0$, in symbols $\gamma_{s*}(0)$, is called the vector field associated with Q. The energy of the longitudinal curves is $E(t) = \int^1_0 <\gamma_{t*}(s),\gamma_{t*}(s)>ds$
The problem: assume the longitudinal curve $\gamma_0(s)$ is a timelike straight line. This mean that $x^i\circ \gamma_0(s) = u^i s + b^i$ and that $<\gamma_{0*}(s),\gamma_{0*}(s)>$ is positive. We know this imply that $\gamma_0(s)$ is an energy-critical curve: $E'(0) = 0$. Assume also that $V$ is normal to $\gamma_0(s)$. I want to prove that $\gamma_0(s)$ is an energy-maximizing curve: $\frac{d^2E}{dt^2}(0) < 0$
I'm having trouble proving this result.