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The causal structure of a spacetime classifies separations and tangent directions as timelike, null or spacelike through its light cones. It determines which events can communicate by causal curves. A null curve is one member of this broader geometric structure.

With the mostly-plus metric signature $(-,+,+,+)$ and coordinates $X^{\mu }=(ct,x,y,z)$, the Minkowski metric is the nondegenerate bilinear formIt is a symmetric bilinear form, but not a positive-definite bilinear form. For example, $U=(1,1,0,0)$ is nonzero yet $U\cdot U=0$, making it a null vector. This does not make the Minkowski metric degenerate: some other vectors have nonzero pairing with $U$.

Applied to infinitesimal displacements, the Minkowski metric givesFor a timelike curve, a clock records $d\tau =\sqrt{-d{s}^2}/c$. For a null curve, $d{s}^2=0$. For spacelike separation, the positive Minkowski interval measures a different kind of separation. A Lorentz transformation preserves this form, expressed in matrix notation by ${\Lambda }^T\eta \Lambda =\eta$, although it need not preserve ordinary Euclidean distance in the four-component coordinate list.

The null condition for a nonzero vector $k$ in a Lorentzian metric is $g(k,k)=0$. Applied to the tangent of a curve it makes the curve a null curve. Nullity alone does not imply the geodesic equation; an additional zero or tangent-parallel covariant acceleration is required.

A null geodesic is both a null curve and a geodesic. In geometric optics, light rays follow null geodesics.

A null pregeodesic has a null tangent and can be parametrized as a null geodesic. In a two-dimensional Lorentzian metric, the acceleration of a regular null curve is orthogonal to its tangent and therefore proportional to it.

For a curve with fixed $v,\theta ,\varphi$, its tangent is proportional to ${\partial }_r$. The Vaidya metric has $g_{rr}=0$, so the curve is a null curve. Moreover, the Christoffel symbols with two lower $r$ indices vanish:because $g_{vr}=1$ is constant and every other $g_{br}$ is zero. Thus the covariant derivative along this tangent gives ${\nabla }_{{\partial }_r}{\partial }_r=0$. The geodesic equation is satisfied with $r$ as an affine parameter; choosing future orientation gives the tangent $k=-{\partial }_r$ and an affine parameter increasing as $-r$. These are the ingoing radial null geodesics.

Let $U$ be the smooth nonzero tangent along a regular null curve, and $a={\nabla }_{U}U$. Metric compatibility gives $g(U,a)=\frac{1}{2}U[g(U,U)]=0$. 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 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.

In Minkowski spacetime with metric $-d{t}^2+d{x}^2+d{y}^2+d{z}^2$, takeIts tangent vector is $(1,-\sin t,\cos t,0)$, with metric tensor square $-1+{\sin }^{2}t+{\cos }^{2}t=0$. Its acceleration is $(0,-\cos t,-\sin t,0)$, nonzero and not proportional to the tangent because its time component is zero. Thus it is a regular null curve which is not a geodesic under any reparametrization.

Take $\alpha >0$ and work in a connected patch with $x\ne 0$. Setting the line element to zero givesThese are the two families of Rindler null geodesics. They remain on the same side of $x=0$ and approach it only at infinite coordinate time. $x=0$ itself is not an additional geodesic within this chart, since the metric is degenerate there.

The paths can also be written $t=\pm {\alpha }^{-1}\log |x/C|$. In two dimensions a null direction has only one independent orthogonal null direction, namely itself, so each smooth null curve is a geodesic after reparametrization. More explicitly, the transformation in part (d) makes these paths straight null lines. The coordinate $t$ is generally not an affine parameter.