Proper Time

For a timelike curve, we can define an invariant proper time as: $\tau =\int \sqrt{-d{s}^2}=\int \sqrt{-{\eta }_{\mu \nu }{\mathring{x}}^{\mu }{\mathring{x}}^{\nu }}d\lambda$ Since ds^2 for a timelike curve is negative, the term inside the square root will be positive.