Physlib.Relativity.Special.ProperTime

For two points $q$ and $p$ in Minkowski spacetime $\text{SpaceTime } d$, the proper time between them is defined as the square root of the Minkowski inner product of their displacement vector, denoted as $\sqrt{\langle p - q, p - q \rangle_{\eta}}$. If the displacement $p - q$ is space-like (resulting in a non-positive value under the square root), the proper time defaults to $0$.

For any two points $q$ and $p$ in $d$-dimensional Minkowski spacetime $\text{SpaceTime } d$, if the displacement vector $p - q$ is time-like, then the proper time between $q$ and $p$ is strictly positive, i.e., $\text{properTime}(q, p) > 0$.

For any two points $q$ and $p$ in Minkowski spacetime $\text{SpaceTime } d$, if the displacement vector $p - q$ has a light-like causal character, then the proper time between $q$ and $p$ is equal to $0$.

In $d$-dimensional Minkowski spacetime $\text{SpaceTime } d$, for any two points $q$ and $p$, if the displacement vector $p - q$ is space-like, then the proper time between $q$ and $p$ is $0$.