Minkowski metric

With the mostly-plus convention, the Minkowski metric is ${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,1,1)$.

For mostly-plus signature, $x^{+}=(x^{D-1}+x^0)/\sqrt{2}$ and $x^{-}=(x^{D-1}-x^0)/\sqrt{2}$ give $d{s}^2=2d{x}^{+}d{x}^{-}+d{x}^{I}d{x}^I$. In this convention raising a plus or minus derivative exchanges the two labels.

The Minkowski norm is the Lorentz-invariant quadratic form $U\cdot U={\eta }_{\mu \nu }{U}^{\mu }{U}^{\nu }$.

The light cone through an event consists of null displacement vectors. In one spatial dimension its two directions satisfy $x=\pm ct$.

The past light cone of an event consists of earlier events connected to it by past-directed null curves. A retarded field at the event depends on sources where their worldlines intersect this cone.

The speed of light $c$ is the invariant speed of every inertial frame and the limiting speed of causal propagation in special relativity.

Null directions are tangent directions of the light cone; their Minkowski squared norm is zero.

A null curve has tangent $u$ satisfying $g(u,u)=0$. It represents the possible spacetime path of a light signal.

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

A radial null geodesic in a spherically symmetric spacetime has zero angular momentum and fixed angular coordinates. Its projection is a null curve of the two-dimensional radial Lorentzian metric.