= Solution
Let $U$ be the smooth nonzero tangent along a regular <null curve>, and $a=\nabla_UU$. Metric compatibility gives $g(U,a)=\tfrac12U[g(U,U)]=0$. In a two-dimensional <Lorentzian manifold>, the <orthogonal complement> of a nonzero <null vector> is its own span; this follows directly in the preceding null <basis>. Therefore $a=\kappa(t)U$.
Consequently \b[every regular <null curve> is a <geodesic> up to reparametrization]. To obtain an <affine parameter> $\lambda$, write $q=dt/d\lambda>0$ and solve $dq/dt+\kappa(t)q=0$. Then $\nabla_{qU}(qU)=q^2(\kappa+q^{-1}dq/dt)U=0$. A solution $q=C\exp(-\int\kappa dt)$ exists locally and along any nonsingular parameter interval. The original parameter need not be affine; the literal assertion of zero covariant <acceleration> for every parameter would be false.
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