On Proper Time in General Relativity

TL;DR

This paper derives the fundamental relation between proper time and proper distance in general relativity, providing a general formula in curved spacetime that simplifies to the special relativistic case in flat spacetime.

Contribution

It presents a derivation of the relation between proper time and proper distance in curved spacetime, extending the understanding from special to general relativity.

Findings

Derived the relation $ds=cd\tau$ in general relativity.

Provided a general relation between $d\tau$ and $dt$ in curved spacetime.

Showed the relation reduces to special relativity in flat spacetime.

Abstract

It is shown how the relation $ds=cd\tau$ between the proper distance $s$ and the proper time $\tau$ is obtained in general relativity. A general relation in curved spacetime between $d\tau$ and $dt$ is given. This relation reduces to the special relativistic one for flat spacetime.