= Sum of future-pointing null vectors
{title2=$(A+B)^2=-2ab(1-\mathbf n\cdot\mathbf m)$}
In <Minkowski spacetime> with <metric signature> $(-,+,+,+)$, write two nonzero future-pointing <null vectors> as $A=a(1,\mathbf n)$ and $B=b(1,\mathbf m)$, where $a,b>0$ and $\mathbf n,\mathbf m$ are <unit vectors>. Their <Lorentzian inner product> gives
$$
(A+B)\cdot(A+B)=-2ab(1-\mathbf n\cdot\mathbf m)\leq0.
$$
Thus their sum is a future-pointing <causal vector>. It is a <null vector> exactly when the spatial directions coincide, and a <timelike vector> otherwise. The positivity of the time components is essential: adding past- and future-pointing <null vectors> can instead produce a <spacelike vector>.
Back to article page
