Proper Length

For a spacelike curve, we can define an invariant line as follows: $l=\int \sqrt{d{s}^2}$ and we define a tangent vector $V^{\mu }=\frac{d{x}^{\mu }}{d\lambda }={\mathring{x}}^{\mu }$ Then, $d{s}^2={\eta }_{\mu \nu }{\mathring{x}}^{\mu }{\mathring{x}}^{\nu }(d\lambda {)}^2$ Note: Proper length is an intrinsic property of the path, i.e., it is independent of both the choice of the parameter and the coordinates.