Minkowski interval ($s^2={\eta }_{\mu \nu }\Delta x^{\mu }\Delta x^{\nu }$)

The squared interval between two events in Minkowski spacetime is the quadratic form of their coordinate difference under the Minkowski metric. In signature $(-,+,+,+)$ it is $-c^2\Delta t^2+|\Delta \mathbf{x}{|}^2$. Lorentz transformations preserve this quadratic form; timelike, spacelike and null separation correspond to negative, positive and zero values.

The opposite metric signature $(+,-,-,-)$ reverses the sign of the Minkowski interval, giving $c^2\Delta t^2-|\Delta \mathbf{x}{|}^2$; the physical classification and proper time stay the same. State the convention before using a sign to identify a separation. With the mostly-plus convention here, a straight clock segment has $s^2=-c^2\Delta {\tau }^2$, while a light signal has $s^2=0$. ^[ref].

Two distinct spacetime events have timelike separation when their displacement has negative Minkowski interval in metric signature $(-,+,+,+)$, equivalently $c|\Delta t|>|\Delta \mathbf{x}|$. A massive observer can connect them along a straight timelike curve. If the later spacetime event has $\Delta t>0$, every Lorentz transformation in the orthochronous Lorentz group keeps it later. The elapsed proper time on the straight inertial segment is $\sqrt{-s^2}/c$.

In the light-second units of the Minkowski spacetime example, $O$ to $A$ has $s^2=(-4^2+3^2){\mathrm{ls}}^2=-7{\mathrm{ls}}^2$. Its straight moving clock therefore records $\sqrt{7}\mathrm{s}$, approximately $2.646\mathrm{s}$, while the coordinate time difference is $4\mathrm{s}$. Coordinate time and proper time answer different questions.

Distinct spacetime events have null separation when their displacement has zero Minkowski interval, equivalently $c|\Delta t|=|\Delta \mathbf{x}|$. They can be joined by a straight vacuum light ray. Their nonzero displacement is a null vector, so the zero squared interval does not mean that the spacetime events coincide. For example, $(t,x)=(3\mathrm{s},3\mathrm{ls})$ is null separated from the origin.